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The Fibonacci sequence is constant-recursive: each element of the sequence is the sum of the previous two.

Hasse diagram of some subclasses of constant-recursive sequences, ordered by inclusion

In mathematics, an infinite sequence of numbers $s_{0},s_{1},s_{2},s_{3},\ldots$ is called constant-recursive if it satisfies an equation of the form

s_{n}=c_{1}s_{n-1}+c_{2}s_{n-2}+\dots +c_{d}s_{n-d},

for all $n\geq d$ , where $c_{i}$ are constants. The equation is called a linear recurrence relation. The concept is also known as a linear recurrence sequence, linear-recursive sequence, linear-recurrent sequence, or a C-finite sequence.^[1]

For example, the Fibonacci sequence

0,1,1,2,3,5,8,13,\ldots

,

is constant-recursive because it satisfies the linear recurrence $F_{n}=F_{n-1}+F_{n-2}$ : each number in the sequence is the sum of the previous two.^[2] Other examples include the power of two sequence $1,2,4,8,16,\ldots$ , where each number is the sum of twice the previous number, and the square number sequence $0,1,4,9,16,25,\ldots$ . All arithmetic progressions, all geometric progressions, and all polynomials are constant-recursive. However, not all sequences are constant-recursive; for example, the factorial sequence $1,1,2,6,24,120,\ldots$ is not constant-recursive.

Constant-recursive sequences are studied in combinatorics and the theory of finite differences. They also arise in algebraic number theory, due to the relation of the sequence to polynomial roots; in the analysis of algorithms, as the running time of simple recursive functions; and in the theory of formal languages, where they count strings up to a given length in a regular language. Constant-recursive sequences are closed under important mathematical operations such as term-wise addition, term-wise multiplication, and Cauchy product.

The Skolem–Mahler–Lech theorem states that the zeros of a constant-recursive sequence have a regularly repeating (eventually periodic) form. The Skolem problem, which asks for an algorithm to determine whether a linear recurrence has at least one zero, is an unsolved problem in mathematics.

Definition

A constant-recursive sequence is any sequence of integers, rational numbers, algebraic numbers, real numbers, or complex numbers $s_{0},s_{1},s_{2},s_{3},\ldots$ (written as $(s_{n})_{n=0}^{\infty }$ as a shorthand) satisfying a formula of the form

s_{n}=c_{1}s_{n-1}+c_{2}s_{n-2}+\dots +c_{d}s_{n-d},

for all $n\geq d,$ for some fixed coefficients $c_{1},c_{2},\dots ,c_{d}$ ranging over the same domain as the sequence (integers, rational numbers, algebraic numbers, real numbers, or complex numbers). The equation is called a linear recurrence with constant coefficients of order d. The order of the sequence is the smallest positive integer $d$ such that the sequence satisfies a recurrence of order d, or $d=0$ for the everywhere-zero sequence.^{[citation needed]}

The definition above allows eventually-periodic sequences such as $1,0,0,0,\ldots$ and $0,1,0,0,\ldots$ . Some authors require that $c_{d}\neq 0$ , which excludes such sequences.^[3]^[4]^[5]

Examples

Selected examples of integer constant-recursive sequences^[6]
Name	Order ( $d$ )	First few values	Recurrence (for $n\geq d$ )	Generating function	OEIS
Zero sequence	0	0, 0, 0, 0, 0, 0, ...	$s_{n}=0$	${\frac {0}{1}}$	A000004
One sequence	1	1, 1, 1, 1, 1, 1, ...	$s_{n}=s_{n-1}$	${\frac {1}{1-x}}$	A000012
Characteristic function of $\{0\}$	1	1, 0, 0, 0, 0, 0, ...	$s_{n}=0$	${\frac {1}{1}}$	A000007
Powers of two	1	1, 2, 4, 8, 16, 32, ...	$s_{n}=2s_{n-1}$	${\frac {1}{1-2x}}$	A000079
Powers of −1	1	1, −1, 1, −1, 1, −1, ...	$s_{n}=-s_{n-1}$	${\frac {1}{1+x}}$	A033999
Characteristic function of $\{1\}$	2	0, 1, 0, 0, 0, 0, ...	$s_{n}=0$	${\frac {x}{1}}$	A063524
Decimal expansion of 1/6	2	1, 6, 6, 6, 6, 6, ...	$s_{n}=s_{n-1}$	${\frac {1+5x}{1-x}}$	A020793
Decimal expansion of 1/11	2	0, 9, 0, 9, 0, 9, ...	$s_{n}=s_{n-2}$	${\frac {9x}{1-x^{2}}}$	A010680
Nonnegative integers	2	0, 1, 2, 3, 4, 5, ...	$s_{n}=2s_{n-1}-s_{n-2}$	${\frac {x}{(1-x)^{2}}}$	A001477
Odd positive integers	2	1, 3, 5, 7, 9, 11, ...	$s_{n}=2s_{n-1}-s_{n-2}$	${\frac {1+x}{(1-x)^{2}}}$	A005408
Fibonacci numbers	2	0, 1, 1, 2, 3, 5, 8, 13, ...	$s_{n}=s_{n-1}+s_{n-2}$	${\frac {x}{1-x-x^{2}}}$	A000045
Lucas numbers	2	2, 1, 3, 4, 7, 11, 18, 29, ...	$s_{n}=s_{n-1}+s_{n-2}$	${\frac {2-x}{1-x-x^{2}}}$	A000032
Pell numbers	2	0, 1, 2, 5, 12, 29, 70, ...	$s_{n}=2s_{n-1}+s_{n-2}$	${\frac {x}{1-2x-x^{2}}}$	A000129
Powers of two interleaved with 0s	2	1, 0, 2, 0, 4, 0, 8, 0, ...	$s_{n}=2s_{n-2}$	${\frac {1}{1-2x^{2}}}$	A077957
Inverse of 6th cyclotomic polynomial	2	1, 1, 0, −1, −1, 0, 1, 1, ...	$s_{n}=s_{n-1}-s_{n-2}$	${\frac {1}{1-x+x^{2}}}$	A010892
Triangular numbers	3	0, 1, 3, 6, 10, 15, 21, ...	$s_{n}=3s_{n-1}-3s_{n-2}+s_{n-3}$	${\frac {x}{(1-x)^{3}}}$	A000217

Fibonacci and Lucas sequences

The sequence 0, 1, 1, 2, 3, 5, 8, 13, ... of Fibonacci numbers is constant-recursive of order 2 because it satisfies the recurrence $F_{n}=F_{n-1}+F_{n-2}$ with $F_{0}=0,F_{1}=1$ . For example, $F_{2}=F_{1}+F_{0}=1+0=1$ and $F_{6}=F_{5}+F_{4}=5+3=8$ . The sequence 2, 1, 3, 4, 7, 11, ... of Lucas numbers satisfies the same recurrence as the Fibonacci sequence but with initial conditions $L_{0}=2$ and $L_{1}=1$ . More generally, every Lucas sequence is constant-recursive of order 2.^[2]

Arithmetic progressions

For any $a$ and any $r\neq 0$ , the arithmetic progression $a,a+r,a+2r,\ldots$ is constant-recursive of order 2, because it satisfies $s_{n}=2s_{n-1}-s_{n-2}$ . Generalizing this, see polynomial sequences below.^{[citation needed]}

Geometric progressions

For any $a\neq 0$ and $r$ , the geometric progression $a,ar,ar^{2},\ldots$ is constant-recursive of order 1, because it satisfies $s_{n}=rs_{n-1}$ . This includes, for example, the sequence 1, 2, 4, 8, 16, ... as well as the rational number sequence ${\textstyle 1,{\frac {1}{2}},{\frac {1}{4}},{\frac {1}{8}},{\frac {1}{16}},...}$ .^{[citation needed]}

Eventually periodic sequences

A sequence that is eventually periodic with period length $\ell$ is constant-recursive, since it satisfies $s_{n}=s_{n-\ell }$ for all $n\geq d$ , where the order $d$ is the length of the initial segment including the first repeating block. Examples of such sequences are 1, 0, 0, 0, ... (order 1) and 1, 6, 6, 6, ... (order 2).^{[citation needed]}

Polynomial sequences

A sequence defined by a polynomial $s_{n}=a_{0}+a_{1}n+a_{2}n^{2}+\cdots +a_{d}n^{d}$ is constant-recursive. The sequence satisfies a recurrence of order $d+1$ (where $d$ is the degree of the polynomial), with coefficients given by the corresponding element of the binomial transform.^[7]^[8] The first few such equations are

s_{n}=1\cdot s_{n-1}

for a degree 0 (that is, constant) polynomial,

s_{n}=2\cdot s_{n-1}-1\cdot s_{n-2}

for a degree 1 or less polynomial,

s_{n}=3\cdot s_{n-1}-3\cdot s_{n-2}+1\cdot s_{n-3}

for a degree 2 or less polynomial, and

s_{n}=4\cdot s_{n-1}-6\cdot s_{n-2}+4\cdot s_{n-3}-1\cdot s_{n-4}

for a degree 3 or less polynomial.

A sequence obeying the order-d equation also obeys all higher order equations. These identities may be proved in a number of ways, including via the theory of finite differences.^[9] Any sequence of $d+1$ integer, real, or complex values can be used as initial conditions for a constant-recursive sequence of order $d+1$ . If the initial conditions lie on a polynomial of degree $d-1$ or less, then the constant-recursive sequence also obeys a lower order equation.

Enumeration of words in a regular language

Let $L$ be a regular language, and let $s_{n}$ be the number of words of length $n$ in $L$ . Then $(s_{n})_{n=0}^{\infty }$ is constant-recursive.^[10] For example, $s_{n}=2^{n}$ for the language of all binary strings, $s_{n}=1$ for the language of all unary strings, and $s_{n}=F_{n+2}$ for the language of all binary strings that do not have two consecutive ones. More generally, any function accepted by a weighted automaton over the unary alphabet $\Sigma =\{a\}$ over the semiring $(\mathbb {R} ,+,\times )$ (which is in fact a ring, and even a field) is constant-recursive.^{[citation needed]}

Other examples

The sequences of Jacobsthal numbers, Padovan numbers, Pell numbers, and Perrin numbers^[2] are constant-recursive.

Non-examples

The factorial sequence $1,1,2,6,24,120,720,\ldots$ is not constant-recursive. More generally, every constant-recursive function is asymptotically bounded by an exponential function (see #Closed-form characterization) and the factorial sequence grows faster than this.

The Catalan sequence $1,1,2,5,14,42,132,\ldots$ is not constant-recursive. This is because the generating function of the Catalan numbers is not a rational function (see #Equivalent definitions).

Equivalent definitions

In terms of matrices

F_{n}={\begin{bmatrix}0&1\end{bmatrix}}{\begin{bmatrix}1&1\\1&0\end{bmatrix}}^{n}{\begin{bmatrix}1\\0\end{bmatrix}}.

Definition of the Fibonacci sequence using matrices.

A sequence $(s_{n})_{n=0}^{\infty }$ is constant-recursive of order less than or equal to $d$ if and only if it can be written as

s_{n}=uA^{n}v

where $u$ is a $1\times d$ vector, $A$ is a $d\times d$ matrix, and $v$ is a $d\times 1$ vector, where the elements come from the same domain (integers, rational numbers, algebraic numbers, real numbers, or complex numbers) as the original sequence. Specifically, $v$ can be taken to be the first $d$ values of the sequence, $A$ the linear transformation that computes $s_{n+1},s_{n+2},\ldots ,s_{n+d}$ from $s_{n},s_{n+1},\ldots ,s_{n+d-1}$ , and $u$ the vector $[0,0,\ldots ,0,1]$ .^[11]

In terms of non-homogeneous linear recurrences

Non-homogeneous	Homogeneous
$s_{n}=1+s_{n-1}$	$s_{n}=2s_{n-1}-s_{n-2}$
$s_{0}=0$	$s_{0}=0;s_{1}=1$

Definition of the sequence of natural numbers

s_{n}=n

, using a non-homogeneous recurrence and the equivalent homogeneous version.

A non-homogeneous linear recurrence is an equation of the form

s_{n}=c_{1}s_{n-1}+c_{2}s_{n-2}+\dots +c_{d}s_{n-d}+c

where $c$ is an additional constant. Any sequence satisfying a non-homogeneous linear recurrence is constant-recursive. This is because subtracting the equation for $s_{n-1}$ from the equation for $s_{n}$ yields a homogeneous recurrence for $s_{n}-s_{n-1}$ , from which we can solve for $s_{n}$ to obtain^{[citation needed]}

{\begin{aligned}s_{n}=&(c_{1}+1)s_{n-1}\\&+(c_{2}-c_{1})s_{n-2}+\dots +(c_{d}-c_{d-1})s_{n-d}\\&-c_{d}s_{n-d-1}.\end{aligned}}

In terms of generating functions

\sum _{n=0}^{\infty }F_{n}x^{n}={\frac {x}{1-x-x^{2}}}.

Definition of the Fibonacci sequence using a generating function.

A sequence is constant-recursive precisely when its generating function

\sum _{n=0}^{\infty }s_{n}x^{n}=s_{0}+s_{1}x^{1}+s_{2}x^{2}+s_{3}x^{3}+\cdots

is a rational function $p(x)\,/\,q(x)$ , where $p$ and $q$ are polynomials and $q(0)=1$ .^[3] Moreover, the order of the sequence is the minimum $d$ such that it has such a form with ${\text{deg }}q(x)\leq d$ and ${\text{deg }}p(x)<d$ .^[12]

The denominator is the polynomial obtained from the auxiliary polynomial by reversing the order of the coefficients, and the numerator is determined by the initial values of the sequence:^[13]^[14]

\sum _{n=0}^{\infty }s_{n}x^{n}={\frac {b_{0}+b_{1}x^{1}+b_{2}x^{2}+\dots +b_{d-1}x^{d-1}}{1-c_{1}x^{1}-c_{2}x^{2}-\dots -c_{d}x^{d}}},

where

b_{n}=s_{n}-c_{1}s_{n-1}-c_{2}s_{n-2}-\dots -c_{d}s_{n-d}.

^[15]

It follows from the above that the denominator $q(x)$ must be a polynomial not divisible by $x$ (and in particular nonzero).

In terms of sequence spaces

\{(an+b)_{n=0}^{\infty }:a,b\in \mathbb {R} \}

2-dimensional vector space of sequences generated by the sequence

s_{n}=n

.

A sequence $(s_{n})_{n=0}^{\infty }$ is constant-recursive if and only if the set of sequences

\left\{(s_{n+r})_{n=0}^{\infty }:r\geq 0\right\}

is contained in a sequence space (vector space of sequences) whose dimension is finite. That is, $(s_{n})_{n=0}^{\infty }$ is contained in a finite-dimensional subspace of $\mathbb {C} ^{\mathbb {N} }$ closed under the left-shift operator.^[16]^[17]

This characterization is because the order- $d$ linear recurrence relation can be understood as a proof of linear dependence between the sequences $(s_{n+r})_{n=0}^{\infty }$ for $r=0,\ldots ,d$ . An extension of this argument shows that the order of the sequence is equal to the dimension of the sequence space generated by $(s_{n+r})_{n=0}^{\infty }$ for all $r$ .^[18]^[17]

Closed-form characterization

F_{n}={\frac {1}{\sqrt {5}}}(1.618\ldots )^{n}-{\frac {1}{\sqrt {5}}}(-0.618\ldots )^{n}

Closed-form characterization of the Fibonacci sequence (Binet's formula)

Constant-recursive sequences admit the following unique closed form characterization using exponential polynomials: every constant-recursive sequence can be written in the form

s_{n}=z_{n}+k_{1}(n)r_{1}^{n}+k_{2}(n)r_{2}^{n}+\cdots +k_{e}(n)r_{e}^{n},

for all $n\geq 0$ , where

The term $z_{n}$ is a sequence which is zero for all $n\geq d$ (where $d$ is the order of the sequence);
The terms $k_{1}(n),k_{2}(n),\ldots ,k_{e}(n)$ are complex polynomials; and
The terms $r_{1},r_{2},\ldots ,r_{k}$ are distinct complex constants.^[19]^[3]

This characterization is exact: every sequence of complex numbers that can be written in the above form is constant-recursive.^[20]

For example, the Fibonacci number $F_{n}$ is written in this form using Binet's formula:^[21]

F_{n}={\frac {1}{\sqrt {5}}}\varphi ^{n}-{\frac {1}{\sqrt {5}}}\psi ^{n},

where $\varphi =(1+{\sqrt {5}})\,/\,2\approx 1.61803\ldots$ is the golden ratio and $\psi =-1\,/\,\varphi$ . These are the roots of the equation $x^{2}-x-1=0$ . In this case, $e=2$ , $z_{n}=0$ for all $n$ , $k_{1}(n)=k_{2}(n)=1\,/\,{\sqrt {5}}$ are both constant polynomials, $r_{1}=\varphi$ , and $r_{2}=\psi$ .

The term $z_{n}$ is only needed when $c_{d}\neq 0$ ; if $c_{d}=0$ then it corrects for the fact that some initial values may be exceptions to the general recurrence. In particular, $z_{n}=0$ for all $n\geq d$ .^{[citation needed]}

The complex numbers $r_{1},\ldots ,r_{n}$ are the roots of the characteristic polynomial of the recurrence:

x^{d}-c_{1}x^{d-1}-\dots -c_{d-1}x-c_{d}

whose coefficients are the same as those of the recurrence.^[22] If the sequence consists of integers or rational numbers, the roots will be algebraic numbers. If the $d$ roots $r_{1},r_{2},\dots ,r_{d}$ are all distinct, then the polynomials $k_{i}(n)$ are all constants, which can be determined from the initial values of the sequence. If the roots of the characteristic polynomial are not distinct, and $r_{i}$ is a root of multiplicity $m$ , then $k_{i}(n)$ in the formula has degree $m-1$ . For instance, if the characteristic polynomial factors as $(x-r)^{3}$ , with the same root r occurring three times, then the $n$ th term is of the form $s_{n}=(a+bn+cn^{2})r^{n}.$ ^[23]^[24]

Closure properties

Examples

The sum of two constant-recursive sequences is also constant-recursive.^[25]^[26] For example, the sum of $s_{n}=2^{n}$ and $t_{n}=n$ is $u_{n}=2^{n}+n$ ( $1,3,6,11,20,\ldots$ ), which satisfies the recurrence $u_{n}=4u_{n-1}-5u_{n-2}+2u_{n-3}$ . The new recurrence can be found by adding the generating functions for each sequence.

Similarly, the product of two constant-recursive sequences is constant-recursive.^[25] For example, the product of $s_{n}=2^{n}$ and $t_{n}=n$ is $u_{n}=n\cdot 2^{n}$ ( $0,2,8,24,64,\ldots$ ), which satisfies the recurrence $u_{n}=4u_{n-1}-4u_{n-2}$ .

The left-shift sequence $u_{n}=s_{n+1}$ and the right-shift sequence $u_{n}=s_{n-1}$ (with $u_{0}=0$ ) are constant-recursive because they satisfy the same recurrence relation. For example, because $s_{n}=2^{n}$ is constant-recursive, so is $u_{n}=2^{n+1}$ .

List of operations

In general, constant-recursive sequences are closed under the following operations, where $s=(s_{n})_{n\in \mathbb {N} },t=(t_{n})_{n\in \mathbb {N} }$ denote constant-recursive sequences, $f(x),g(x)$ are their generating functions, and $d,e$ are their orders, respectively.^[27]

Operations on constant-recursive sequences
Operation	Definition	Requirement	Generating function equivalent	Order
Term-wise sum $s+t$	$(s+t)_{n}=s_{n}+t_{n}$	—	$f(x)+g(x)$	$\leq d+e$ ^[25]
Term-wise product $s\cdot t$	$(s\cdot t)_{n}=s_{n}\cdot t_{n}$	—	${\frac {1}{2\pi i}}\int _{\gamma }{\frac {f(\zeta )}{\zeta }}g\left({\frac {x}{\zeta }}\right)\;\mathrm {d} \zeta$ ^[28]^[29]	$\leq d\cdot e$ ^[11]^[25]
Cauchy product $s*t$	$(s*t)_{n}=\sum _{i=0}^{n}s_{i}t_{n-i}$	—	$f(x)g(x)$	$\leq d+e$ ^[27]
Left shift $Ls$	$(Ls)_{n}=s_{n+1}$	—	${\frac {f(x)-s_{0}}{x}}$	$\leq d$ ^[27]
Right shift $Rs$	$(Rs)_{n}={\begin{cases}s_{n-1}&n\geq 1\\0&n=0\end{cases}}$	—	$xf(x)$	$\leq d+1$ ^[27]
Cauchy inverse $s^{(-1)}$	$(s^{(-1)})_{n}=\sum _{{i_{1}+\dots +i_{k}=n} \atop {i_{1},\ldots ,i_{k}\neq 0}}(-1)^{k}s_{i_{1}}s_{i_{2}}\cdots s_{i_{k}}$	$s_{0}=1$	${\frac {1}{f(x)}}$	$\leq d+1$ ^[27]
Kleene star $s^{(*)}$	$(s^{(*)})_{n}=\sum _{{i_{1}+\dots +i_{k}=n} \atop {i_{1},\ldots ,i_{k}\neq 0}}s_{i_{1}}s_{i_{2}}\cdots s_{i_{k}}$	$s_{0}=0$	${\frac {1}{1-f(x)}}$	$\leq d+1$ ^[27]

The closure under term-wise addition and multiplication follows from the closed-form characterization in terms of exponential polynomials. The closure under Cauchy product follows from the generating function characterization.^[27] The requirement $s_{0}=1$ for Cauchy inverse is necessary for the case of integer sequences, but can be replaced by $s_{0}\neq 0$ if the sequence is over any field (rational, algebraic, real, or complex numbers).^[27]

Behavior

Unsolved problem in mathematics:

Is there an algorithm to test whether a constant-recursive sequence has a zero?

(more unsolved problems in mathematics)

Zeros

Despite satisfying a simple local formula, a constant-recursive sequence can exhibit complicated global behavior. Define a zero of a constant-recursive sequence to be a nonnegative integer $n$ such that $s_{n}=0$ . The Skolem–Mahler–Lech theorem states that the zeros of the sequence are eventually repeating: there exists constants $M$ and $N$ such that for all $n>M$ , $s_{n}=0$ if and only if $s_{n+N}=0$ . This result holds for a constant-recursive sequence over the complex numbers, or more generally, over any field of characteristic zero.^[30]

Decision problems

The pattern of zeros in a constant-recursive sequence can also be investigated from the perspective of computability theory. To do so, the description of the sequence $s_{n}$ must be given a finite description; this can be done if the sequence is over the integers or rational numbers, or even over the algebraic numbers.^[11] Given such an encoding for sequences $s_{n}$ , the following problems can be studied:

Notable decision problems
Problem	Description	Status^[11]^[31]
Existence of a zero (Skolem problem)	On input $(s_{n})_{n=0}^{\infty }$ , is $s_{n}=0$ for some $n$ ?	Open
Infinitely many zeros	On input $(s_{n})_{n=0}^{\infty }$ , is $s_{n}=0$ for infinitely many $n$ ?	Decidable
Eventually all zero	On input $(s_{n})_{n=0}^{\infty }$ , is $s_{n}=0$ for all sufficiently large $n$ ?	Decidable
Positivity	On input $(s_{n})_{n=0}^{\infty }$ , is $s_{n}>0$ for all $n$ ?	Open
Eventual positivity	On input $(s_{n})_{n=0}^{\infty }$ , is $s_{n}>0$ for all sufficiently large $n$ ?	Open

Because the square of a constant-recursive sequence $s_{n}^{2}$ is still constant-recursive (see closure properties), the existence-of-a-zero problem in the table above reduces to positivity, and infinitely-many-zeros reduces to eventual positivity. Other problems also reduce to those in the above table: for example, whether $s_{n}=c$ for some $n$ reduces to existence-of-a-zero for the sequence $s_{n}-c$ . As a second example, for sequences in the real numbers, weak positivity (is $s_{n}\geq 0$ for all $n$ ?) reduces to positivity of the sequence $-s_{n}$ (because the answer must be negated, this is a Turing reduction).

The Skolem-Mahler-Lech theorem would provide answers to some of these questions, except that its proof is non-constructive. It states that for all $n>M$ , the zeros are repeating; however, the value of $M$ is not known to be computable, so this does not lead to a solution to the existence-of-a-zero problem.^[11] On the other hand, the exact pattern which repeats after $n>M$ is computable.^[11]^[32] This is why the infinitely-many-zeros problem is decidable: just determine if the infinitely-repeating pattern is empty.

Decidability results are known when the order of a sequence is restricted to be small. For example, the Skolem problem is decidable for sequences of order up to 4,^[11] and for a restricted set of sequences up to order 7.^[31]

Generalizations

A D-finite or holonomic sequence is a natural generalization where the coefficients of the recurrence are allowed to be polynomial functions of $n$ rather than constants.^[33]

A $k$ -regular sequence satisfies a linear recurrences with constant coefficients, but the recurrences take a different form. Rather than $s_{n}$ being a linear combination of $s_{m}$ for some integers $m$ that are close to $n$ , each term $s_{n}$ in a $k$ -regular sequence is a linear combination of $s_{m}$ for some integers $m$ whose base- $k$ representations are close to that of $n$ .^[34] Constant-recursive sequences can be thought of as $1$ -regular sequences, where the base-1 representation of $n$ consists of $n$ copies of the digit $1$ .^{[citation needed]}

Notes

^ Kauers & Paule 2010, p. 63.
^ ^a ^b ^c Kauers & Paule 2010, p. 70.
^ ^a ^b ^c Stanley 2011, p. 464.
^ Kauers & Paule 2010, p. 66.
^ Halava, Vesa; Harju, Tero; Hirvensalo, Mika; Karhumäki, Juhani (2005), Skolem's Problem – On the Border between Decidability and Undecidability, p. 1, CiteSeerX 10.1.1.155.2606
^ "Index to OEIS: Section Rec - OeisWiki". oeis.org. Retrieved 2024-04-18.
^ Boyadzhiev, Boyad (2012). "Close Encounters with the Stirling Numbers of the Second Kind" (PDF). Math. Mag. 85 (4): 252–266. arXiv:1806.09468. doi:10.4169/math.mag.85.4.252. S2CID 115176876.
^ Riordan, John (1964). "Inverse Relations and Combinatorial Identities". The American Mathematical Monthly. 71 (5): 485–498. doi:10.1080/00029890.1964.11992269. ISSN 0002-9890.
^ Jordan, Charles; Jordán, Károly (1965). Calculus of Finite Differences. American Mathematical Soc. pp. 9–11. ISBN 978-0-8284-0033-6. See formula on p.9, top.
^ Kauers & Paule 2010, p. 81.
^ ^a ^b ^c ^d ^e ^f ^g Ouaknine, Joël; Worrell, James (2012), "Decision problems for linear recurrence sequences", Reachability Problems: 6th International Workshop, RP 2012, Bordeaux, France, September 17–19, 2012, Proceedings, Lecture Notes in Computer Science, vol. 7550, Heidelberg: Springer-Verlag, pp. 21–28, doi:10.1007/978-3-642-33512-9_3, ISBN 978-3-642-33511-2, MR 3040104.
^ Stanley 2011, pp. 464–465.
^ Martino, Ivan; Martino, Luca (2013-11-14). "On the variety of linear recurrences and numerical semigroups". Semigroup Forum. 88 (3): 569–574. arXiv:1207.0111. doi:10.1007/s00233-013-9551-2. ISSN 0037-1912. S2CID 119625519.
^ Kauers & Paule 2010, p. 74.
^ Stanley 2011, pp. 468–469.
^ Kauers & Paule 2010, p. 67.
^ ^a ^b Stanley 2011, p. 465.
^ Kauers & Paule 2010, p. 69.
^ Brousseau 1971, pp. 28–34, Lesson 5.
^ Kauers & Paule 2010, pp. 68–70.
^ Brousseau 1971, p. 16, Lesson 3.
^ Brousseau 1971, p. 28, Lesson 5.
^ Greene, Daniel H.; Knuth, Donald E. (1982), "2.1.1 Constant coefficients – A) Homogeneous equations", Mathematics for the Analysis of Algorithms (2nd ed.), Birkhäuser, p. 17.
^ Brousseau 1971, pp. 29–31, Lesson 5.
^ ^a ^b ^c ^d Kauers & Paule 2010, p. 71.
^ Brousseau 1971, p. 37, Lesson 6.
^ ^a ^b ^c ^d ^e ^f ^g ^h Stanley 2011, pp. 471.
^ Pohlen, Timo (2009). "The Hadamard product and universal power series" (PDF). University of Trier (Doctoral Dissertation): 36–37.
^ See Hadamard product (series) and Parseval's theorem.
^ Lech, C. (1953), "A Note on Recurring Series", Arkiv för Matematik, 2 (5): 417–421, Bibcode:1953ArM.....2..417L, doi:10.1007/bf02590997
^ ^a ^b Lipton, Richard; Luca, Florian; Nieuwveld, Joris; Ouaknine, Joël; Purser, David; Worrell, James (2022-08-04). "On the Skolem Problem and the Skolem Conjecture". Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science. LICS '22. New York, NY, USA: Association for Computing Machinery. pp. 1–9. doi:10.1145/3531130.3533328. ISBN 978-1-4503-9351-5.
^ Berstel, Jean; Mignotte, Maurice (1976). "Deux propriétés décidables des suites récurrentes linéaires". Bulletin de la Société Mathématique de France (in French). 104: 175–184. doi:10.24033/bsmf.1823.
^ Stanley, Richard P (1980). "Differentiably finite power series". European journal of combinatorics. 1 (2): 175–188.
^ Allouche, Jean-Paul; Shallit, Jeffrey (1992). "The ring of k-regular sequences". Theoretical Computer Science. 98 (2): 163–197.

References

Brousseau, Alfred (1971). Linear Recursion and Fibonacci Sequences. Fibonacci Association.
Kauers, Manuel; Paule, Peter (2010). The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates. Springer Vienna. p. 66. ISBN 978-3-7091-0444-6.
Stanley, Richard P. (2011). Enumerative Combinatorics (PDF). Vol. 1 (2 ed.). Cambridge studies in advanced mathematics.

External links

"OEIS Index Rec". OEIS index to a few thousand examples of linear recurrences, sorted by order (number of terms) and signature (vector of values of the constant coefficients)

[FOOTNOTEKauersPaule201063-1] Kauers & Paule 2010, p. 63.

[FOOTNOTEKauersPaule201070-2] Kauers & Paule 2010, p. 70.

[FOOTNOTEStanley2011464-3] Stanley 2011, p. 464.

[FOOTNOTEKauersPaule201066-4] Kauers & Paule 2010, p. 66.

[Border-5] Halava, Vesa; Harju, Tero; Hirvensalo, Mika; Karhumäki, Juhani (2005), Skolem's Problem – On the Border between Decidability and Undecidability, p. 1, CiteSeerX 10.1.1.155.2606

[6] "Index to OEIS: Section Rec - OeisWiki". oeis.org. Retrieved 2024-04-18.

[7] Boyadzhiev, Boyad (2012). "Close Encounters with the Stirling Numbers of the Second Kind" (PDF). Math. Mag. 85 (4): 252–266. arXiv:1806.09468. doi:10.4169/math.mag.85.4.252. S2CID 115176876.

[8] Riordan, John (1964). "Inverse Relations and Combinatorial Identities". The American Mathematical Monthly. 71 (5): 485–498. doi:10.1080/00029890.1964.11992269. ISSN 0002-9890.

[9] Jordan, Charles; Jordán, Károly (1965). Calculus of Finite Differences. American Mathematical Soc. pp. 9–11. ISBN 978-0-8284-0033-6. See formula on p.9, top.

[FOOTNOTEKauersPaule201081-10] Kauers & Paule 2010, p. 81.

[ow-11] ^ ^a ^b ^c ^d ^e ^f ^g Ouaknine, Joël; Worrell, James (2012), "Decision problems for linear recurrence sequences", Reachability Problems: 6th International Workshop, RP 2012, Bordeaux, France, September 17–19, 2012, Proceedings, Lecture Notes in Computer Science, vol. 7550, Heidelberg: Springer-Verlag, pp. 21–28, doi:10.1007/978-3-642-33512-9_3, ISBN 978-3-642-33511-2, MR 3040104.

[FOOTNOTEStanley2011464–465-12] Stanley 2011, pp. 464–465.

[13] Martino, Ivan; Martino, Luca (2013-11-14). "On the variety of linear recurrences and numerical semigroups". Semigroup Forum. 88 (3): 569–574. arXiv:1207.0111. doi:10.1007/s00233-013-9551-2. ISSN 0037-1912. S2CID 119625519.

[FOOTNOTEKauersPaule201074-14] Kauers & Paule 2010, p. 74.

[FOOTNOTEStanley2011468–469-15] Stanley 2011, pp. 468–469.

[FOOTNOTEKauersPaule201067-16] Kauers & Paule 2010, p. 67.

[FOOTNOTEStanley2011465-17] Stanley 2011, p. 465.

[FOOTNOTEKauersPaule201069-18] Kauers & Paule 2010, p. 69.

[FOOTNOTEBrousseau197128–34Lesson_5-19] Brousseau 1971, pp. 28–34, Lesson 5.

[FOOTNOTEKauersPaule201068–70-20] Kauers & Paule 2010, pp. 68–70.

[FOOTNOTEBrousseau197116Lesson_3-21] Brousseau 1971, p. 16, Lesson 3.

[FOOTNOTEBrousseau197128Lesson_5-22] Brousseau 1971, p. 28, Lesson 5.

[23] Greene, Daniel H.; Knuth, Donald E. (1982), "2.1.1 Constant coefficients – A) Homogeneous equations", Mathematics for the Analysis of Algorithms (2nd ed.), Birkhäuser, p. 17.

[FOOTNOTEBrousseau197129–31Lesson_5-24] Brousseau 1971, pp. 29–31, Lesson 5.

[FOOTNOTEKauersPaule201071-25] Kauers & Paule 2010, p. 71.

[FOOTNOTEBrousseau197137Lesson_6-26] Brousseau 1971, p. 37, Lesson 6.

[FOOTNOTEStanley2011471-27] ^ ^a ^b ^c ^d ^e ^f ^g ^h Stanley 2011, pp. 471.

[28] Pohlen, Timo (2009). "The Hadamard product and universal power series" (PDF). University of Trier (Doctoral Dissertation): 36–37.

[29] See Hadamard product (series) and Parseval's theorem.

[30] Lech, C. (1953), "A Note on Recurring Series", Arkiv för Matematik, 2 (5): 417–421, Bibcode:1953ArM.....2..417L, doi:10.1007/bf02590997

[LICS22-31] Lipton, Richard; Luca, Florian; Nieuwveld, Joris; Ouaknine, Joël; Purser, David; Worrell, James (2022-08-04). "On the Skolem Problem and the Skolem Conjecture". Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science. LICS '22. New York, NY, USA: Association for Computing Machinery. pp. 1–9. doi:10.1145/3531130.3533328. ISBN 978-1-4503-9351-5.

[32] Berstel, Jean; Mignotte, Maurice (1976). "Deux propriétés décidables des suites récurrentes linéaires". Bulletin de la Société Mathématique de France (in French). 104: 175–184. doi:10.24033/bsmf.1823.

[33] Stanley, Richard P (1980). "Differentiably finite power series". European journal of combinatorics. 1 (2): 175–188.

[34] Allouche, Jean-Paul; Shallit, Jeffrey (1992). "The ring of k-regular sequences". Theoretical Computer Science. 98 (2): 163–197.

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Revision as of 03:54, 19 April 2024

Definition

Examples

Fibonacci and Lucas sequences

Arithmetic progressions

Geometric progressions

Eventually periodic sequences

Polynomial sequences

Enumeration of words in a regular language

Other examples

Non-examples

Equivalent definitions

In terms of matrices

In terms of non-homogeneous linear recurrences

In terms of generating functions

In terms of sequence spaces

Closed-form characterization

Closure properties

Examples

List of operations

Behavior

Zeros

Decision problems

Generalizations

Notes

References

External links

Leave a Reply

@@ Line 1: / Line 1: @@
 {{short description|Infinite sequence of numbers satisfying a linear equation}}
-{{Further information|Linear difference equation}}
-[[File:Fibonacci sequence.jpg|thumb|300px|right|The Fibonacci sequence is constant-recursive: each element of the sequence is the sum of the previous two.]] In [[mathematics]] and [[theoretical computer science]], a '''constant-recursive sequence''' or '''linear recurrence sequence''' is an [[sequence|infinite sequence]] of [[number]]s in which each number in the sequence is equal to a [[linear combination]] of one or more of its immediate predecessors.
+[[File:Fibonacci sequence.jpg|thumb|350px|right|The Fibonacci sequence is constant-recursive: each element of the sequence is the sum of the previous two.]]
+[[File:Constant-recursive-sequences.svg|thumb|350px|right|[[Hasse diagram]] of some subclasses of constant-recursive sequences, ordered by inclusion]]
+In [[mathematics]], an [[sequence|infinite sequence]] of [[number]]s <math>s_0, s_1, s_2, s_3, \ldots</math> is called '''constant-recursive''' if it satisfies an equation of the form
-The most famous example of a constant-recursive sequence is the [[Fibonacci number|Fibonacci sequence]], in which each number is the sum of the previous two; formally, we say that it satisfies the [[Linear difference equation|linear recurrence relation]] <math>F(n) = F(n-1) + F(n-2)</math>, where <math>F(n)</math> is the <math>n</math>th Fibonacci number. Generalizing this idea, a constant-recursive sequence <math>s(n)</math> satisfies a formula of the form
-:<math display=block>s(n) = c_1 s(n-1) + c_2 s(n-2) + \dots + c_d s(n-d),</math>
+:<math display=block>s_n = c_1 s_{n-1} + c_2 s_{n-2} + \dots + c_d s_{n-d},</math>
+for all <math>n \ge d</math>, where <math>c_i</math> are [[constant (mathematics)|constant]]s. The equation is called a [[linear recurrence with constant coefficients|linear recurrence relation]].
-where <math>c_i</math> are [[constant]]s. In addition to the Fibonacci sequence, constant-recursive sequences include many other well-known sequences, such as [[arithmetic progression]]s, [[geometric progression]]s, and [[polynomial]]s.
+The concept is also known as a '''linear recurrence sequence''', '''linear-recursive sequence''', '''linear-recurrent sequence''', or a '''C-finite sequence'''.{{sfn|Kauers|Paule|2010|p=63}}
+For example, the [[Fibonacci sequence]]
-Constant-recursive sequences arise in many areas, including [[combinatorics]], due to the [[Combinatorial principles#Recurrence relation|combinatorial interpretation of recurrence relations]]; in [[algebraic number theory]], due to the relation of the sequence to the roots of a polynomial; in the theory of [[finite difference]]s; in the [[analysis of algorithms]] in the running time of simple [[Recursion (computer science)|recursive function]]s; and in other settings, for example in [[formal language theory]], where they count the number of strings in a [[regular language]]. Constant-recursive sequences are closed under important mathematical operations such as [[pointwise]] [[addition]], pointwise [[multiplication]], and [[Cauchy product]]. A seminal result is the [[Skolem–Mahler–Lech theorem]], which states that the [[zeros of a function|zeros]] of a constant-recursive sequence have a regularly-repeating (eventually-periodic) form. On the other hand, the [[Skolem problem]], which asks for [[algorithm]] to determine whether a linear recurrence has at least one zero, remains one of the [[List of unsolved problems in mathematics|unsolved problems in mathematics]].
+:<math>0, 1, 1, 2, 3, 5, 8, 13, \ldots</math>,
+is constant-recursive because it satisfies the linear recurrence <math>F_n = F_{n-1} + F_{n-2}</math>: each number in the sequence is the sum of the previous two.{{sfn|Kauers|Paule|2010|p=70}}
+Other examples include the [[power of two]] sequence <math>1, 2, 4, 8, 16, \ldots</math>, where each number is the sum of twice the previous number, and the [[square number]] sequence <math>0, 1, 4, 9, 16, 25, \ldots</math>. All [[arithmetic progression]]s, all [[geometric progression]]s, and all [[polynomial]]s are constant-recursive. However, not all sequences are constant-recursive; for example, the [[factorial]] sequence <math>1, 1, 2, 6, 24, 120, \ldots</math> is not constant-recursive.
+Constant-recursive sequences are studied in [[combinatorics]] and the theory of [[finite difference]]s. They also arise in [[algebraic number theory]], due to the relation of the sequence to [[polynomial root]]s; in the [[analysis of algorithms]], as the running time of simple [[Recursion (computer science)|recursive function]]s; and in the theory of [[formal language]]s, where they count strings up to a given length in a [[regular language]]. Constant-recursive sequences are [[closure (mathematics)|closed]] under important mathematical operations such as [[pointwise|term-wise]] [[addition]], term-wise [[multiplication]], and [[Cauchy product]].
+The [[Skolem–Mahler–Lech theorem]] states that the [[Zero of a function|zeros]] of a constant-recursive sequence have a regularly repeating (eventually periodic) form. The [[Skolem problem]], which asks for an [[algorithm]] to determine whether a linear recurrence has at least one zero, is an [[List of unsolved problems in mathematics|unsolved problem in mathematics]].
 ==Definition==
-An order-''d'' '''homogeneous [[linear difference equation|linear recurrence relation]]'''  is an equation of the form
-:<math>s(n) = c_1 s(n-1) + c_2 s(n-2) + \dots + c_d s(n-d),</math>
-where the ''d'' coefficients <math>c_1, c_2, \dots, c_d</math> are [[Constant (mathematics)|constants]]. The coefficients are taken to range over the [[integer]]s, [[rational number]]s, [[algebraic number]]s, [[real number]]s, or [[complex number]]s.
+A '''constant-recursive sequence''' is any sequence of [[integer]]s, [[rational number]]s, [[algebraic number]]s, [[real number]]s, or [[complex number]]s <math>s_0, s_1, s_2, s_3, \ldots</math> (written as <math>(s_n)_{n=0}^\infty</math> as a shorthand) satisfying a formula of the form
-A sequence <math>s(0), s(1), s(2), \dots</math> (ranging over the same domain as the coefficients) is a '''constant-recursive sequence''' if there is an order-''d'' homogeneous linear recurrence with constant coefficients that it satisfies for all <math>n \geq d</math>. Note that this definition allows eventually-periodic sequences such as <math>1, 0, 0, 0, \ldots</math> and <math>0, 1, 0, 0, \ldots</math>, which are disallowed by some authors.<ref>{{Cite book|last=Kauers|first=Manuel|url=https://books.google.com/books?id=BPeODAEACAAJ&newbks=0&hl=en|title=The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates|last2=Paule|first2=Peter|date=2010-12-01|publisher=Springer Vienna|isbn=978-3-7091-0444-6|language=en}}</ref> These sequences can be excluded by requiring that <math>c_d \ne 0</math>.
+<math display=block>s_n = c_1 s_{n-1} + c_2 s_{n-2} + \dots + c_d s_{n-d},</math>
-The '''order''' of a constant-recursive sequence is the smallest <math>d \ge 1</math> such that the sequence satisfies an order-''d'' homogeneous linear recurrence, or <math>d = 0</math> for the everywhere-zero sequence.
+for all <math>n \ge d,</math> for some fixed [[coefficient]]s <math>c_1, c_2, \dots, c_d</math> ranging over the same domain as the sequence (integers, rational numbers, algebraic numbers, real numbers, or complex numbers).
-==Equivalent definitions==
+The equation is called a [[linear recurrence with constant coefficients]] of order ''d''.
+The ''order'' of the sequence is the smallest positive integer <math>d</math> such that the sequence satisfies a recurrence of order ''d'', or <math>d = 0</math> for the everywhere-zero sequence.
+{{cn|date=April 2024}}
+The definition above allows eventually-[[periodic sequence|periodic]] sequences such as <math>1, 0, 0, 0, \ldots</math> and <math>0, 1, 0, 0, \ldots</math>. Some authors require that <math>c_d \ne 0</math>, which excludes such sequences.{{sfn|Stanley|2011|p=464}}{{sfn|Kauers|Paule|2010|p=66}}<ref name=Border>{{Citation|last1=Halava|first1=Vesa|title=Skolem's Problem – On the Border between Decidability and Undecidability|date=2005|last2=Harju|first2=Tero|last3=Hirvensalo|first3=Mika|last4=Karhumäki|first4=Juhani|citeseerx=10.1.1.155.2606|page=1}}</ref>
-===In terms of vector spaces===
+==Examples==
-A sequence <math>s(n)_{n \geq 0}</math> is constant-recursive if and only if the set of sequences
-:<math>\{s(n + r)_{n \geq 0} : r \geq 0\}</math>
-is contained in a [[vector space]] whose dimension is finite, i.e. a finite-dimensional subspace of <math>\mathbb{C}^\mathbb{N}</math> closed under the [[Shift operator#Sequences|left-shift operator]]. This is because the order-<math>d</math> linear recurrence relation can be understood as a [[Linear independence#Definition|proof of linear dependence]] between the vectors <math>s(n+r)_{n \geq 0}</math> for <math>r=0, \ldots, d</math>. An extension of this argument shows that the order of the sequence corresponds to the dimension of the vector space generated by <math>s(n+r)_{n \geq 0}</math> for all <math>r</math>.
+{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
-===In terms of non-homogeneous linear recurrences===
+|+ Selected examples of integer constant-recursive sequences<ref>{{Cite web |title=Index to OEIS: Section Rec - OeisWiki |url=https://oeis.org/wiki/Index_to_OEIS:_Section_Rec |access-date=2024-04-18 |website=oeis.org}}</ref>
+! Name !! Order (<math>d</math>&hairsp;&hairsp;) !! First few values !! Recurrence (for <math>n \ge d</math>&hairsp;) !! [[Generating function]] !! [[OEIS]]
+|-
+| Zero sequence || 0 || 0, 0, 0, 0, 0, 0, ... || <math>s_n = 0</math> || <math>\frac{0}{1}</math> || {{OEIS link|A000004}}
+|-
+| One sequence || 1 || 1, 1, 1, 1, 1, 1, ... || <math>s_n = s_{n-1}</math> || <math>\frac{1}{1-x}</math> || {{OEIS link|A000012}}
+|-
+| [[Indicator function|Characteristic function]] of <math>\{0\}</math> || 1 || 1, 0, 0, 0, 0, 0,  ... || <math>s_n = 0</math> || <math>\frac{1}{1}</math> || {{OEIS link|A000007}}
+|-
+| [[Powers of two]] || 1 || 1, 2, 4, 8, 16, 32, ... || <math>s_n = 2 s_{n-1}</math> || <math>\frac{1}{1-2x}</math> || {{OEIS link|A000079}}
+|-
+| Powers of −1|| 1 || 1, −1, 1, −1, 1, −1, ... || <math>s_n = -s_{n-1}</math> || <math>\frac{1}{1+x}</math> || {{OEIS link|A033999}}
+|-
+| Characteristic function of <math>\{1\}</math> || 2 || 0, 1, 0, 0, 0, 0, ... || <math>s_n = 0</math> || <math>\frac{x}{1}</math> || {{OEIS link|A063524}}
+|-
+| [[Decimal representation|Decimal expansion]] of 1/6 || 2 || 1, 6, 6, 6, 6, 6, ... || <math>s_n = s_{n-1}</math> || <math>\frac{1 + 5x}{1 - x}</math> || {{OEIS link|A020793}}
+|-
+| Decimal expansion of 1/11 || 2 || 0, 9, 0, 9, 0, 9, ... || <math>s_n = s_{n-2}</math> || <math>\frac{9x}{1-x^2}</math> || {{OEIS link|A010680}}
+|-
+| [[Natural number|Nonnegative integers]] || 2 || 0, 1, 2, 3, 4, 5, ... || <math>s_n = 2s_{n-1} - s_{n-2}</math> || <math>\frac{x}{(1-x)^2}</math> || {{OEIS link|A001477}}
+|-
+| [[Parity (mathematics)|Odd]] positive integers || 2 || 1, 3, 5, 7, 9, 11, ... || <math>s_n = 2s_{n-1} - s_{n-2}</math> || <math>\frac{1+x}{(1-x)^2}</math> || {{OEIS link|A005408}}
+|-
+| [[Fibonacci number]]s || 2 || 0, 1, 1, 2, 3, 5, 8, 13, ... || <math>s_n = s_{n-1} + s_{n-2}</math> || <math>\frac{x}{1-x-x^2}</math> || {{OEIS link|A000045}}
+|-
+| [[Lucas number]]s || 2 || 2, 1, 3, 4, 7, 11, 18, 29, ... || <math>s_n = s_{n-1} + s_{n-2}</math> || <math>\frac{2-x}{1-x-x^2}</math> || {{OEIS link|A000032}}
+|-
+| [[Pell number]]s || 2 || 0, 1, 2, 5, 12, 29, 70, ... || <math>s_n = 2s_{n-1} + s_{n-2}</math> || <math>\frac{x}{1-2x-x^2}</math> || {{OEIS link|A000129}}
+|-
+| Powers of two [[Interleave sequence|interleaved with]] 0s || 2 || 1, 0, 2, 0, 4, 0, 8, 0, ... || <math>s_n = 2 s_{n-2}</math> || <math>\frac{1}{1-2x^2}</math> || {{OEIS link|A077957}}
+|-
+| Inverse of 6th [[cyclotomic polynomial]] || 2 || 1, 1, 0, −1, −1, 0, 1, 1, ... || <math>s_n = s_{n-1} - s_{n-2}</math> || <math>\frac{1}{1-x+x^2}</math> || {{OEIS link|A010892}}
+|-
+| [[Triangular number]]s || 3 || 0, 1, 3, 6, 10, 15, 21, ... || <math>s_n = 3s_{n-1} - 3s_{n-2} + s_{n-3}</math> || <math>\frac{x}{(1-x)^3}</math> || {{OEIS link|A000217}}
+|}
+===Fibonacci and Lucas sequences===
-A sequence is constant-recursive if and only if it satisfies a [[linear difference equation|''non-homogenous'' linear recurrence]] of the form
+The sequence 0, 1, 1, 2, 3, 5, 8, 13, ... of [[Fibonacci number]]s is constant-recursive of order 2 because it satisfies the recurrence <math>F_n = F_{n-1} + F_{n-2}</math> with <math>F_0 = 0, F_1 = 1</math>. For example, <math>F_2 = F_1 + F_0 = 1 + 0 = 1</math> and <math>F_6 = F_5 + F_4 = 5 + 3 = 8</math>. The sequence 2, 1, 3, 4, 7, 11, ... of [[Lucas number]]s satisfies the same recurrence as the Fibonacci sequence but with initial conditions <math>L_0 = 2</math> and <math>L_1 = 1</math>. More generally, every [[Lucas sequence]] is constant-recursive of order 2.{{sfn|Kauers|Paule|2010|p=70}}
-:<math>s(n) = c_1 s(n-1) + c_2 s(n-2) + \dots + c_d s(n-d) + c</math>
-for all <math>n \ge d</math>, where <math>c</math> is an additional constant. This is because the non-homogeneous recurrence
-can be solved for <math>c</math> to obtain
-:<math>c = s(n) - c_1 s(n-1) - c_2 s(n-2) - \dots - c_d s(n-d).</math>
-Substituting this into the equation
-:<math>s(n+1) = c_1 s(n) + c_2 s(n-1) + \dots + c_d s(n+1-d) + c</math>
-shows that <math>s(n)</math> satisfies the homogeneous recurrence
-:<math>s(n+1) = (c_1 + 1) s(n) + (c_2 - c_1) s(n-1) + \dots + (c_d  - c_{d-1}) s(n+1-d) - c_d s(n-d)</math>
-of order <math>d + 1</math>.
+===Arithmetic progressions===
-==Examples==
+For any <math>a</math> and any <math>r \ne 0</math>, the [[arithmetic progression]] <math>a, a+r, a+2r, \ldots</math> is constant-recursive of order 2, because it satisfies <math>s_n = 2s_{n-1} - s_{n-2}</math>. Generalizing this, see [[Constant-recursive sequence#Polynomial sequences|polynomial sequences]] below.
+{{cn|date=April 2024}}
-===Fibonacci sequence===
+===Geometric progressions===
+For any <math>a \ne 0</math> and <math>r</math>, the [[geometric progression]] <math>a, a r, a r^2, \ldots</math> is constant-recursive of order 1, because it satisfies <math>s_n = r s_{n-1}</math>. This includes, for example, the sequence 1, 2, 4, 8, 16, ... as well as the rational number sequence <math display=inline>1, \frac12, \frac14, \frac18, \frac{1}{16}, ...</math>.
-The sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... of [[Fibonacci number]]s satisfies the recurrence
+{{cn|date=April 2024}}
-:<math>F_n = F_{n-1}+F_{n-2}</math>
+===Eventually periodic sequences===
-with [[initial condition]]s
+A sequence that is eventually periodic with period length <math>\ell</math> is constant-recursive, since it satisfies <math>s_n = s_{n-\ell}</math> for all <math>n \geq d</math>, where the order <math>d</math> is the length of the initial segment including the first repeating block. Examples of such sequences are 1, 0, 0, 0, ... (order 1) and 1, 6, 6, 6, ... (order 2).
-:<math>F_0 = 0</math>
+{{cn|date=April 2024}}
-:<math>F_1 = 1.</math>
+===Polynomial sequences===
-Explicitly, the recurrence yields the values
+A sequence defined by a polynomial <math>s_n = a_0 + a_1 n + a_2 n^2 + \cdots + a_d n^d</math> is constant-recursive. The sequence satisfies a recurrence of order <math>d + 1</math> (where <math>d</math> is the [[degree of a polynomial|degree]] of the polynomial), with coefficients given by the corresponding element of the [[binomial transform]].<ref>{{Cite journal|last=Boyadzhiev|first=Boyad|date=2012|title=Close Encounters with the Stirling Numbers of the Second Kind|url=https://www.maa.org/sites/default/files/pdf/upload_library/2/Boyadzhiev-2013.pdf|journal=Math. Mag.|volume=85|issue=4|pages=252–266|doi=10.4169/math.mag.85.4.252|arxiv=1806.09468|s2cid=115176876}}</ref><ref>{{Cite journal |last=Riordan |first=John |date=1964 |title=Inverse Relations and Combinatorial Identities |url=https://www.tandfonline.com/doi/full/10.1080/00029890.1964.11992269 |journal=The American Mathematical Monthly |language=en |volume=71 |issue=5 |pages=485–498 |doi=10.1080/00029890.1964.11992269 |issn=0002-9890}}</ref> The first few such equations are
-:<math>F_2 = F_1 + F_0 = 1</math>
-:<math>F_3 = F_2 + F_1 = 2</math>
-:<math>F_4 = F_3 + F_2 = 3</math>
-:<math>F_5 = F_4 + F_3 = 5</math>
+: <math>s_n = 1 \cdot s_{n-1}</math> for a degree 0 (that is, constant) polynomial,
-etc.
+: <math>s_n = 2\cdot s_{n-1} - 1\cdot s_{n-2}</math> for a degree 1 or less polynomial,
+: <math>s_n = 3\cdot s_{n-1} - 3\cdot s_{n-2} + 1\cdot s_{n-3}</math> for a degree 2 or less polynomial, and
+: <math>s_n = 4\cdot s_{n-1} - 6\cdot s_{n-2} + 4\cdot s_{n-3} - 1\cdot s_{n-4}</math> for a degree 3 or less polynomial.
+A sequence obeying the order-''d'' equation also obeys all higher order equations. These identities may be [[mathematical proof|proved]] in a number of ways, including via the theory of [[finite difference]]s.<ref>{{Cite book|last1=Jordan|first1=Charles|url=https://books.google.com/books?id=3RfZOsDAyQsC&dq=theory+of+finite+differences&pg=PA1|title=Calculus of Finite Differences|last2=Jordán|first2=Károly|date=1965|publisher=American Mathematical Soc.|isbn=978-0-8284-0033-6|language=en|pages=9–11}} See formula on p.9, top.</ref>
-===Lucas sequences===
+Any sequence of <math>d + 1</math> integer, real, or complex values can be used as initial conditions for a constant-recursive sequence of order <math>d + 1</math>. If the initial conditions lie on a polynomial of degree <math>d - 1</math> or less, then the constant-recursive sequence also obeys a lower order equation.
-The sequence 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, ... {{OEIS|id=A000032|name=Lucas numbers beginning at 2: L(n) = L(n-1) + L(n-2), L(0) = 2, L(1) = 1}} of [[Lucas number]]s satisfies the same recurrence as the Fibonacci sequence but with initial conditions
-:<math>L_0 = 2</math>
-:<math>L_1 = 1.</math>
+===Enumeration of words in a regular language===
-More generally, every [[Lucas sequence]] is a constant-recursive sequence.
+Let <math>L</math> be a [[regular language]], and let <math>s_n</math> be the number of words of length <math>n</math> in <math>L</math>. Then <math>(s_n)_{n=0}^\infty</math> is constant-recursive.{{sfn|Kauers|Paule|2010|p=81}} For example, <math>s_n = 2^n</math> for the language of all binary strings, <math>s_n = 1</math> for the language of all unary strings, and <math>s_n = F_{n+2}</math> for the language of all binary strings that do not have two consecutive ones. More generally, any function accepted by a [[weighted automaton]] over the unary alphabet <math>\Sigma = \{a\}</math> over the [[semiring]] <math>(\mathbb{R}, +, \times)</math> (which is in fact a [[ring (mathematics)|ring]], and even a [[field (mathematics)|field]]) is constant-recursive.
-===Geometric sequences===
+{{cn|date=April 2024}}
-The [[geometric sequence]] <math>a, a r, a r^2, \dots</math> is constant-recursive, since it satisfies the recurrence <math>s(n) = r s(n - 1)</math> for all <math>n \geq 1</math>.
+===Other examples===
-===Eventually periodic sequences===
-A sequence that is eventually periodic with period length <math>\ell</math> is constant-recursive, since it satisfies <math>s(n) = s(n - \ell)</math> for all <math>n \geq d</math> for some <math>d</math>.
+The sequences of [[Jacobsthal number]]s, [[Padovan number]]s, [[Pell number]]s, and [[Perrin number]]s{{sfn|Kauers|Paule|2010|p=70}} are constant-recursive.
-===Polynomial sequences===
-For any polynomial ''s''(''n''), the sequence of its values is a constant-recursive sequence.  If the degree of the polynomial is ''d'', the sequence satisfies a recurrence of order <math>d + 1</math>, with coefficients given by the corresponding element of the [[binomial transform]].<ref>{{Cite journal|last=Boyadzhiev|first=Boyad|date=2012|title=Close Encounters with the Stirling Numbers of the Second Kind|url=https://www.maa.org/sites/default/files/pdf/upload_library/2/Boyadzhiev-2013.pdf|journal=Math. Mag.|volume=85|pages=252–266}}</ref>  The first few such equations are
+===Non-examples===
-:<math> s(n) = 1 \cdot s(n-1) </math> for a degree 0 (that is, constant) polynomial,
-:<math> s(n) = 2\cdot s(n-1) - 1\cdot s(n-2) </math> for a degree 1 or less polynomial,
-:<math> s(n) = 3\cdot s(n-1) - 3\cdot s(n-2) + 1\cdot s(n-3) </math> for a degree 2 or less polynomial, and
-:<math> s(n) = 4\cdot s(n-1) - 6\cdot s(n-2) + 4\cdot s(n-3) - 1\cdot s(n-4)</math> for a degree 3 or less polynomial.
+The [[factorial]] sequence <math>1, 1, 2, 6, 24, 120, 720, \ldots</math> is not constant-recursive. More generally, every constant-recursive function is asymptotically bounded by an [[exponential function]] (see [[#Closed-form characterization]]) and the factorial sequence grows faster than this.
-A sequence obeying the order-''d'' equation also obeys all higher order equations. These identities may be proved in a number of ways, including via the theory of [[finite differences]].{{cn|date=May 2018}} Each individual equation may also be verified by substituting the degree-''d'' polynomial
+The [[Catalan sequence]] <math>1, 1, 2, 5, 14, 42, 132, \ldots</math> is not constant-recursive. This is because the [[Catalan number#First proof|generating function of the Catalan numbers]] is not a [[rational function]] (see [[#Equivalent definitions]]).
-:<math>s(x)=\sum_{k=0}^d a_k x^k,</math>
+==Equivalent definitions==
-where the coefficients <math>a_k</math> are symbolic. Any sequence of <math>d + 1</math> integer, real, or complex values can be used as initial conditions for a constant-recursive sequence of order <math>d + 1</math>. If the initial conditions lie on a polynomial of degree <math>d - 1</math> or less, then the constant-recursive sequence also obeys a lower order equation.
-===Enumeration of words in a regular language===
+===In terms of matrices===
+{{main|Companion matrix}}
-Let <math>L</math> be a [[regular language]], and let <math>s(n)</math> be the number of words of length <math>n</math> in <math>L</math>. Then <math>s(n)_{n \geq 0}</math> is constant-recursive.
+{{float begin|width=195px|side=right}}
+|-style="text-align:center;"
+| <math>F_n = \begin{bmatrix}0 & 1\end{bmatrix}
+\begin{bmatrix}1 & 1 \\ 1 & 0\end{bmatrix}^n
+\begin{bmatrix}1 \\ 0\end{bmatrix}.</math>
+{{float end|Definition of the Fibonacci sequence using matrices.}}
+A sequence <math>(s_n)_{n=0}^\infty</math> is constant-recursive of order less than or equal to <math>d</math> if and only if it can be written as
-===Other examples===
+:<math>s_n = u A^n v</math>
+where <math>u</math> is a <math>1 \times d</math> vector, <math>A</math> is a <math>d \times d</math> [[matrix (mathematics)|matrix]], and <math>v</math> is a <math>d \times 1</math> vector, where the elements come from the same domain (integers, rational numbers, algebraic numbers, real numbers, or complex numbers) as the original sequence. Specifically, <math>v</math> can be taken to be the first <math>d</math> values of the sequence, <math>A</math> the [[linear transformation]] that computes <math>s_{n+1}, s_{n+2}, \ldots, s_{n+d}</math> from <math>s_n, s_{n+1}, \ldots, s_{n+d-1}</math>, and <math>u</math> the vector <math>[0, 0, \ldots, 0, 1]</math>.<ref name="ow"/>
+===In terms of non-homogeneous linear recurrences===
-The sequences of [[Jacobsthal number]]s, [[Padovan number]]s, [[Pell number]]s, and [[Perrin number]]s are constant-recursive.
+{{main|Linear recurrence with constant coefficients#Conversion to homogeneous form}}
-== Characterization in terms of exponential polynomials ==
+{{float begin|width=250px|side=right}}
-The [[characteristic equation (calculus)|characteristic polynomial]] (or "auxiliary polynomial") of the recurrence is the polynomial
+|- class="wikitable"
-:<math>x^d - c_1 x^{d-1} - \dots - c_{d-1} x - c_d</math>
+! Non-homogeneous !! Homogeneous
-whose coefficients are the same as those of the recurrence.
+|- align = "center"
-If <math>c_d \ne 0</math>, the ''n''th term <math>s(n)</math> of a constant-recursive sequence can be written in terms of the [[root of a polynomial | roots]] of its characteristic polynomial.
+| <math>s_n = 1 + s_{n-1}</math>
-If the ''d'' roots <math>r_1, r_2, \dots, r_d</math> are all distinct, then the ''n''th term of the sequence is
-:<math>s(n) = k_1 r_1^n + k_2 r_2^n + \dots + k_d r_d^n</math>
+| <math>s_n = 2s_{n-1} - s_{n-2}</math>
+|- align = "center"
-where the coefficients ''k<sub>i</sub>'' are constants that can be determined from the initial conditions.
+| <math>s_0 = 0</math>
+| <math>s_0 = 0; s_1 = 1</math>
+{{float end|Definition of the sequence of [[natural number]]s <math>s_n = n</math>, using a non-homogeneous recurrence and the equivalent homogeneous version.}}
+A '''non-homogeneous linear recurrence''' is an equation of the form
-For the Fibonacci sequence, the characteristic polynomial is <math>x^2 - x - 1</math>, whose roots <math>\phi = \frac{1 + \sqrt{5}}{2}</math> and <math>\bar{\phi} = \frac{1 - \sqrt{5}}{2}</math> appear in [[Binet's formula]]
-:<math>F_n = \frac{\phi^n-\bar{\phi}^n}{\sqrt 5}.</math>
+:<math>s_n = c_1 s_{n-1} + c_2 s_{n-2} + \dots + c_d s_{n-d} + c</math>
+where <math>c</math> is an additional constant. Any sequence satisfying a non-homogeneous linear recurrence is constant-recursive. This is because subtracting the equation for <math>s_{n-1}</math> from the equation for <math>s_n</math> yields a homogeneous recurrence for <math>s_n - s_{n-1}</math>, from which we can solve for <math>s_n</math> to obtain{{cn|date=April 2024}}
+:<math>\begin{align}s_n = &(c_1 + 1) s_{n-1} \\ &+ (c_2 - c_1) s_{n-2} + \dots + (c_d  - c_{d-1}) s_{n-d} \\&- c_d s_{n-d-1}.\end{align}</math>
+=== In terms of generating functions ===
-More generally, if a root ''r'' of the characteristic polynomial has multiplicity ''m'', then the term <math>r^n</math> is multiplied by a degree-<math>(m-1)</math> polynomial in ''n''. That is, let <math>r_1, \dots, r_e</math> be the distinct roots of the characteristic polynomial. Then
-:<math>s(n) = k_1(n) r_1^n + k_2(n) r_2^n + \dots + k_e(n) r_e^n</math>
-where <math>k_i(n)</math> is a polynomial of degree <math>m_i - 1</math>.
-For instance, if the characteristic polynomial factors as <math>(x-r)^3</math>, with the same root ''r'' occurring three times, then the ''n''th term is of the form
-:<math>s(n) = (a + b n + c n^2) r^n.</math><ref>{{citation|contribution=2.1.1 Constant coefficients – A) Homogeneous equations|title=Mathematics for the Analysis of Algorithms|first1=Daniel H.|last1=Greene|first2=Donald E.|last2=Knuth|author2-link=Donald Knuth| edition=2nd| publisher=Birkhäuser |page=17 | year=1982}}.</ref>
+{{float begin|width=180px|side=right}}
-Conversely, if there are polynomials <math>k_i(n)</math> such that
+|-style="text-align:center;"
-:<math>s(n) = k_1(n) r_1^n + k_2(n) r_2^n + \dots + k_e(n) r_e^n,</math>
-then <math>s(n)_{n \geq 0}</math> is constant-recursive.
+|<math>\sum_{n = 0}^\infty F_n x^n = \frac{x}{1-x-x^2}.</math>
+{{float end|caption=Definition of the Fibonacci sequence using a generating function.}}
-== Characterization in terms of rational generating functions ==
 A sequence is constant-recursive precisely when its [[generating function]]
-:<math>\sum_{n \geq 0} s(n) x^n = s(0) + s(1) x^1 + s(2) x^2 + s(3) x^3 + \cdots</math>
+:<math>\sum_{n = 0}^\infty s_n x^n = s_0 + s_1 x^1 + s_2 x^2 + s_3 x^3 + \cdots</math>
+is a rational function <math>p(x) \,/\, q(x)</math>, where <math>p</math> and <math>q</math> are polynomials and <math>q(0) = 1</math>.{{sfn|Stanley|2011|p=464}}
-is a [[rational function]]. The denominator is the polynomial obtained from the auxiliary polynomial by reversing the order of the coefficients, and the numerator is determined by the initial values of the sequence.<ref>{{Cite journal|title = On the variety of linear recurrences and numerical semigroups|journal = [[Semigroup Forum]]|date = 2013-11-14|issn = 0037-1912|pages = 569–574|volume = 88|issue = 3|doi = 10.1007/s00233-013-9551-2|language = en|first = Ivan|last = Martino|first2 = Luca|last2 = Martino|arxiv = 1207.0111}}</ref>
+Moreover, the order of the sequence is the minimum <math>d</math> such that it has such a form with <math>\text{deg } q(x) \le d</math> and <math>\text{deg } p(x) < d</math>.{{sfn|Stanley|2011|pp=464-465}}
+The denominator is the polynomial obtained from the auxiliary polynomial by [[reciprocal polynomial|reversing the order of the coefficients]], and the numerator is determined by the initial values of the sequence:<ref>{{Cite journal|title = On the variety of linear recurrences and numerical semigroups|journal = [[Semigroup Forum]]|date = 2013-11-14|issn = 0037-1912|pages = 569–574|volume = 88|issue = 3|doi = 10.1007/s00233-013-9551-2|language = en|first1 = Ivan|last1 = Martino|first2 = Luca|last2 = Martino|arxiv = 1207.0111|s2cid = 119625519}}</ref>{{sfn|Kauers|Paule|2010|p=74}}
-The generating function of the Fibonacci sequence is
-:<math>\frac{x}{1-x-x^2}.</math>
+:<math>\sum_{n = 0}^\infty s_n x^n = \frac{b_0 + b_1 x^1 + b_2 x^2 + \dots + b_{d-1} x^{d-1}}{1 - c_1 x^1 - c_2 x^2 - \dots - c_d x^d},</math>
-In general, multiplying a generating function by the polynomial
-:<math>1 - c_1 x^1 - c_2 x^2 - \dots - c_d x^d</math>
-yields a series
-:<math>\left(s(0) + s(1) x^1 + s(2) x^2 + \cdots\right) \left(1 - c_1 x^1 - c_2 x^2 - \dots - c_d x^d \right) = \left(b_0 + b_1 x^1 + b_2 x^2 + \cdots\right),</math>
 where
-:<math>b_n = s(n) - c_1 s(n-1) - c_2 s(n-2) - \dots - c_d s(n-d).</math>
+:<math>b_n = s_n - c_1 s_{n-1} - c_2 s_{n-2} - \dots - c_d s_{n-d}.</math>{{sfn|Stanley|2011|pp=468-469}}
+It follows from the above that the denominator <math>q(x)</math> must be a polynomial not divisible by <math>x</math> (and in particular nonzero).
-If <math>s(n)</math> satisfies the recurrence relation
-:<math>s(n) = c_1 s(n-1) + c_2 s(n-2) + \dots + c_d s(n-d),</math>
-then <math>b_n = 0</math> for all <math>n \geq d</math>.  In other words,
-:<math>\left(s(0) + s(1) x^1 + s(2) x^2 + \cdots\right) \left(1 - c_1 x^1 - c_2 x^2 - \dots - c_d x^d \right) = \left(b_0 + b_1 x^1 + b_2 x^2 + \dots + b_{d-1} x^{d-1}\right),</math>
-so we obtain the rational function
-:<math>\sum_{n \geq 0} s(n) x^n = \frac{b_0 + b_1 x^1 + b_2 x^2 + \dots + b_{d-1} x^{d-1}}{1 - c_1 x^1 - c_2 x^2 - \dots - c_d x^d}.</math>
+===In terms of sequence spaces===
-In the special case of a periodic sequence satisfying <math>s(n) = s(n-d)</math> for <math>n \geq d</math>, the generating function is
-:<math>\begin{align}\frac{s(0) + s(1) x^1 + \dots + s(d-1) x^{d-1}}{1-x^d} =& \left(s(0) + s(1) x^1 + \dots + s(d-1) x^{d-1}\right) + \left(s(0) + s(1) x^1 + \dots + s(d-1) x^{d-1}\right) x^d + {}\\& \left(s(0) + s(1) x^1 + \dots + s(d-1) x^{d-1}\right) x^{2d} + \cdots\end{align}</math>
-by expanding the geometric series.
+{{float begin|width=175px|side=right}}
-The [[Catalan number#First proof|generating function of the Catalan numbers]] is not a rational function, which implies that the [[Catalan numbers]] do not satisfy a linear recurrence with constant coefficients.
+|-align=center
+|<math>\{(a n + b)_{n=0}^\infty : a, b \in \mathbb{R}\}</math>
+{{float end|2-dimensional [[sequence space|vector space of sequences]] generated by the sequence <math>s_n = n</math>.}}
+A sequence <math>(s_n)_{n=0}^\infty</math> is constant-recursive if and only if the set of sequences
+:<math>\left\{(s_{n+r})_{n=0}^\infty : r \geq 0\right\}</math>
+is contained in a [[sequence space]] ([[vector space]] of sequences) whose [[Dimension (vector space)|dimension]] is finite. That is, <math>(s_n)_{n=0}^\infty</math> is contained in a finite-dimensional [[Linear subspace|subspace]] of <math>\mathbb{C}^\mathbb{N}</math> [[Closure (mathematics)|closed]] under the [[Shift operator#Sequences|left-shift operator]].{{sfn|Kauers|Paule|2010|p=67}}{{sfn|Stanley|2011|p=465}}
+This characterization is because the order-<math>d</math> linear recurrence relation can be understood as a [[Linear independence#Definition|proof of linear dependence]] between the sequences <math>(s_{n+r})_{n=0}^\infty</math> for <math>r=0, \ldots, d</math>. An extension of this argument shows that the order of the sequence is equal to the dimension of the sequence space generated by <math>(s_{n+r})_{n=0}^\infty</math> for all <math>r</math>.{{sfn|Kauers|Paule|2010|p=69}}{{sfn|Stanley|2011|p=465}}
+== Closed-form characterization ==
+{{for|a derivation of the closed form|Linear recurrence with constant coefficients#General solution}}
+{{float_begin|side=right|width=300px}}
+|-align=center
+|<math>F_n = \frac{1}{\sqrt{5}}(1.618\ldots)^n - \frac{1}{\sqrt{5}}(-0.618\ldots)^n</math>
+{{float_end|Closed-form characterization of the Fibonacci sequence ([[Fibonacci number#Binet's formula|Binet's formula]])}}
+Constant-recursive sequences admit the following unique [[Closed-form expression|closed form]] characterization using [[exponential polynomial]]s: every constant-recursive sequence can be written in the form
+:<math>s_n = z_n + k_1(n) r_1^n + k_2(n) r_2^n + \cdots + k_e(n) r_e^n,</math>
+for all <math>n \ge 0</math>, where
+* The term <math>z_n</math> is a sequence which is zero for all <math>n \ge d</math> (where <math>d</math> is the order of the sequence);
+* The terms <math>k_1(n), k_2(n), \ldots, k_e(n)</math> are complex polynomials; and
+* The terms <math>r_1, r_2, \ldots, r_k</math> are distinct complex constants.{{sfn|Brousseau|1971|pp=28-34|loc=Lesson 5}}{{sfn|Stanley|2011|p=464}}
+This characterization is exact: every sequence of complex numbers that can be written in the above form is constant-recursive.{{sfn|Kauers|Paule|2010|pp=68-70}}
+For example, the Fibonacci number <math>F_n</math> is written in this form using [[Fibonacci number#Binet's formula|Binet's formula]]:{{sfn|Brousseau|1971|p=16|loc=Lesson 3}}
+:<math>F_n = \frac{1}{\sqrt{5}}\varphi^n - \frac{1}{\sqrt{5}}\psi^n,</math>
+where <math>\varphi = (1 + \sqrt{5}) \,/\, 2 \approx 1.61803\ldots</math> is the [[golden ratio]] and <math>\psi = -1 \,/\, \varphi</math>. These are the roots of the equation <math>x^2 - x - 1 = 0</math>. In this case, <math>e=2</math>, <math>z_n = 0</math> for all <math>n</math>, <math>k_1(n) = k_2(n) = 1 \,/\, \sqrt{5}</math> are both constant polynomials, <math>r_1 = \varphi</math>, and <math>r_2 = \psi</math>.
+The term <math>z_n</math> is only needed when <math>c_d\ne 0</math>; if <math>c_d = 0</math> then it corrects for the fact that some initial values may be exceptions to the general recurrence. In particular, <math>z_n = 0</math> for all <math>n \ge d</math>.{{cn|date=April 2024}}
+The complex numbers <math>r_1, \ldots, r_n</math> are the roots of the [[characteristic equation (calculus)|characteristic polynomial]] of the recurrence:
+:<math>x^d - c_1 x^{d-1} - \dots - c_{d-1} x - c_d</math>
+whose coefficients are the same as those of the recurrence.{{sfn|Brousseau|1971|p=28|loc=Lesson 5}}
+If the sequence consists of integers or rational numbers, the roots will be [[algebraic number]]s.
+If the <math>d</math> roots <math>r_1, r_2, \dots, r_d</math> are all distinct, then the polynomials <math>k_i(n)</math> are all constants, which can be determined from the initial values of the sequence.
+If the roots of the characteristic polynomial are not distinct, and <math>r_i</math> is a root of [[Multiplicity (mathematics)|multiplicity]] <math>m</math>, then <math>k_i(n)</math> in the formula has degree <math>m - 1</math>. For instance, if the characteristic polynomial factors as <math>(x-r)^3</math>, with the same root ''r'' occurring three times, then the <math>n</math>th term is of the form <math>s_n = (a + b n + c n^2) r^n.</math><ref>{{citation|contribution=2.1.1 Constant coefficients – A) Homogeneous equations|title=Mathematics for the Analysis of Algorithms|first1=Daniel H.|last1=Greene|first2=Donald E.|last2=Knuth|author2-link=Donald Knuth| edition=2nd| publisher=Birkhäuser |page=17 | year=1982}}.</ref>{{sfn|Brousseau|1971|pp=29-31|loc=Lesson 5}}
 ==Closure properties==
-The termwise addition or multiplication of two constant-recursive sequences is again constant-recursive.  This follows from the characterization in terms of exponential polynomials.
+===Examples===
-The [[Cauchy product]] of two constant-recursive sequences is constant-recursive.  This follows from the characterization in terms of rational generating functions.
+The sum of two constant-recursive sequences is also constant-recursive.{{sfn|Kauers|Paule|2010|p=71}}{{sfn|Brousseau|1971|p=37|loc=Lesson 6}} For example, the sum of <math>s_n = 2^n</math> and <math>t_n = n</math> is <math>u_n = 2^n + n</math> (<math>1, 3, 6, 11, 20, \ldots</math>), which satisfies the recurrence <math>u_n = 4u_{n-1} - 5u_{n-2} + 2u_{n-3}</math>. The new recurrence can be found by adding the generating functions for each sequence.
+Similarly, the product of two constant-recursive sequences is constant-recursive.{{sfn|Kauers|Paule|2010|p=71}} For example, the product of <math>s_n = 2^n</math> and <math>t_n = n</math> is <math>u_n = n \cdot 2^n</math> (<math>0, 2, 8, 24, 64, \ldots</math>), which satisfies the recurrence <math>u_n = 4 u_{n-1} - 4 u_{n-2}</math>.
+The left-shift sequence <math>u_n = s_{n + 1}</math> and the right-shift sequence <math>u_n = s_{n - 1}</math> (with <math>u_0 = 0</math>) are constant-recursive because they satisfy the same recurrence relation. For example, because <math>s_n = 2^n</math> is constant-recursive, so is <math>u_n = 2^{n + 1}</math>.
+===List of operations===
+In general, constant-recursive sequences are [[Closure (mathematics)|closed]] under the following operations, where <math>s = (s_n)_{n \in \mathbb{N}}, t = (t_n)_{n \in \mathbb{N}}</math> denote constant-recursive sequences, <math>f(x), g(x)</math> are their generating functions, and <math>d, e</math> are their orders, respectively.{{sfn|Stanley|2011|pp=471}}
+{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
+|+ Operations on constant-recursive sequences
+! Operation !! Definition !! Requirement !! Generating function equivalent || Order
+|-
+| Term-wise sum <math>s + t</math> || <math>(s + t)_n = s_n + t_n</math> || — || <math>f(x) + g(x)</math> || <math>\le d + e</math>{{sfn|Kauers|Paule|2010|p=71}}
+|-
+| Term-wise product <math>s \cdot t</math> || <math>(s \cdot t)_n = s_n \cdot t_n</math> || — || <math>\frac{1}{2 \pi i} \int_\gamma \frac{f(\zeta)}{\zeta} g\left(\frac{x}{\zeta}\right) \; \mathrm{d}\zeta</math><ref>{{Cite journal |last=Pohlen |first=Timo |date=2009 |title=The Hadamard product and universal power series |url=https://ubt.opus.hbz-nrw.de/opus45-ubtr/frontdoor/deliver/index/docId/327/file/Dissertation.pdf |pp=36-37 |journal=University of Trier (Doctoral Dissertation)}}</ref><ref>See [[Hadamard product (series)]] and [[Parseval's theorem]].</ref> || <math> \le d \cdot e </math><ref name="ow"/>{{sfn|Kauers|Paule|2010|p=71}}
+|-
+| [[Cauchy product]] <math>s * t</math> || <math>(s * t)_n = \sum_{i=0}^n s_i t_{n-i}</math> || — || <math>f(x) g(x)</math> || <math>\le d + e</math>{{sfn|Stanley|2011|pp=471}}
+|-
+| Left shift <math>Ls</math> || <math>(Ls)_n = s_{n+1}</math> || — || <math>\frac{f(x) - s_0}{x}</math> || <math>\le d</math>{{sfn|Stanley|2011|pp=471}}
+|-
+| Right shift <math>Rs</math> || <math>(Rs)_n = \begin{cases}s_{n-1} &n \ge 1\\0 &n = 0\end{cases}</math> || — || <math>x f(x)</math> || <math>\le d + 1</math>{{sfn|Stanley|2011|pp=471}}
+|-
+| [[Formal power series#Multiplicative inverse|Cauchy inverse]] <math>s^{(-1)}</math> || <math>(s^{(-1)})_n = \sum_{{i_1 + \dots + i_k = n} \atop {i_1, \ldots, i_k \ne 0}} (-1)^k s_{i_1} s_{i_2} \cdots s_{i_k}</math> || <math>s_0 = 1</math> || <math>\frac{1}{f(x)}</math> || <math>\le d + 1</math>{{sfn|Stanley|2011|pp=471}}
+|-
+| [[Kleene star]] <math>s^{(*)}</math> || <math>(s^{(*)})_n = \sum_{{i_1 + \dots + i_k = n} \atop {i_1, \ldots, i_k \ne 0}} s_{i_1} s_{i_2} \cdots s_{i_k}</math> || <math> s_0 = 0</math> || <math>\frac{1}{1 - f(x)}</math> || <math>\le d + 1</math>{{sfn|Stanley|2011|pp=471}}
+|}
+The closure under term-wise addition and multiplication follows from the closed-form characterization in terms of exponential polynomials. The closure under Cauchy product follows from the generating function characterization.{{sfn|Stanley|2011|pp=471}} The requirement <math>s_0 = 1</math> for Cauchy inverse is necessary for the case of integer sequences, but can be replaced by <math>s_0 \ne 0</math> if the sequence is over any [[field (mathematics)|field]] (rational, algebraic, real, or complex numbers).{{sfn|Stanley|2011|pp=471}}
+<!--
+TBD: closure operations characterization:
+The constant-recursive sequences can be characterized as the smallest set of sequences containing the constants (in integer, rational, algebraic, or complex numbers) closed under pointwise addition, right-shift, Cauchy product, and either Cauchy inverse or Kleene star.
+A characterization involving product might also be possible.
+Need to include a citation for these statements.
+-->
+==Behavior==
+{{main|Skolem–Mahler–Lech theorem|Skolem problem}}
+{{Unsolved|mathematics|Is there an algorithm to test whether a constant-recursive sequence has a zero?}}
+===Zeros===
+Despite satisfying a simple local formula, a constant-recursive sequence can exhibit complicated global behavior. Define a ''zero'' of a constant-recursive sequence to be a nonnegative integer <math>n</math> such that <math>s_n = 0</math>. The Skolem–Mahler–Lech theorem states that the zeros of the sequence are eventually repeating: there exists constants <math>M</math> and <math>N</math> such that for all <math>n > M</math>, <math>s_n = 0</math> if and only if <math>s_{n+N} = 0</math>. This result holds for a constant-recursive sequence over the complex numbers, or more generally, over any [[Field (mathematics)|field]] of [[Characteristic (algebra)|characteristic]] zero.<ref>{{citation|last=Lech|first=C.|title=A Note on Recurring Series|journal=Arkiv för Matematik|volume=2|pages=417–421|year=1953|issue=5|doi=10.1007/bf02590997|bibcode=1953ArM.....2..417L |doi-access=free}}</ref>
+===Decision problems===
+The pattern of zeros in a constant-recursive sequence can also be investigated from the perspective of [[computability theory]]. To do so, the description of the sequence <math>s_n</math> must be given a [[String (computer science)|finite description]]; this can be done if the sequence is over the integers or rational numbers, or even over the algebraic numbers.<ref name="ow">{{citation
+ | last1 = Ouaknine | first1 = Joël
+ | last2 = Worrell | first2 = James
+ | contribution = Decision problems for linear recurrence sequences
+ | doi = 10.1007/978-3-642-33512-9_3
+ | mr = 3040104
+ | pages = 21–28
+ | publisher = Springer-Verlag | location = Heidelberg
+ | series = Lecture Notes in Computer Science
+ | title = Reachability Problems: 6th International Workshop, RP 2012, Bordeaux, France, September 17–19, 2012, Proceedings
+ | volume = 7550
+ | year = 2012| isbn = 978-3-642-33511-2
+ }}.</ref>
+Given such an encoding for sequences <math>s_n</math>, the following problems can be studied:
+{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
+|+ Notable decision problems
+! Problem !! Description !! Status<ref name="ow"/><ref name=LICS22>{{Cite book |last1=Lipton |first1=Richard |last2=Luca |first2=Florian |last3=Nieuwveld |first3=Joris |last4=Ouaknine |first4=Joël |last5=Purser |first5=David |last6=Worrell |first6=James |chapter=On the Skolem Problem and the Skolem Conjecture |date=2022-08-04 |title=Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science |chapter-url=https://dl.acm.org/doi/10.1145/3531130.3533328 |series=LICS '22 |location=New York, NY, USA |publisher=Association for Computing Machinery |pages=1–9 |doi=10.1145/3531130.3533328 |isbn=978-1-4503-9351-5}}</ref>
+|-
+| Existence of a zero ([[Skolem problem]])
+  || On input <math>(s_n)_{n=0}^\infty</math>, is <math>s_n = 0</math> for some <math>n</math>?
+  || Open
+|-
+| Infinitely many zeros
+  || On input <math>(s_n)_{n=0}^\infty</math>, is <math>s_n = 0</math> for infinitely many <math>n</math>?
+  || Decidable
+|-
+| Eventually all zero
+  || On input <math>(s_n)_{n=0}^\infty</math>, is <math>s_n = 0</math> for all [[Eventually (mathematics)|sufficiently large]] <math>n</math>?
+  || Decidable
+|-
+| Positivity
+  || On input <math>(s_n)_{n=0}^\infty</math>, is <math>s_n > 0</math> for all <math>n</math>?
+  || Open
+|-
+| Eventual positivity
+  || On input <math>(s_n)_{n=0}^\infty</math>, is <math>s_n > 0</math> for all [[Eventually (mathematics)|sufficiently large]] <math>n</math>?
+  || Open
+|}
+Because the square of a constant-recursive sequence <math>s_n^2</math> is still constant-recursive (see [[#Closure properties|closure properties]]), the existence-of-a-zero problem in the table above [[Reduction (recursion theory)|reduces to]] positivity, and infinitely-many-zeros reduces to eventual positivity. Other problems also reduce to those in the above table: for example, whether <math>s_n = c</math> for some <math>n</math> reduces to existence-of-a-zero for the sequence <math>s_n - c</math>. As a second example, for sequences in the real numbers, ''weak'' positivity (is <math>s_n \ge 0</math> for all <math>n</math>?) reduces to positivity of the sequence <math>-s_n</math> (because the answer must be negated, this is a [[Turing reduction]]).
+The Skolem-Mahler-Lech theorem would provide answers to some of these questions, except that its proof is [[Constructive proof|non-constructive]]. It states that for all <math>n > M</math>, the zeros are repeating; however, the value of <math>M</math> is not known to be computable, so this does not lead to a solution to the existence-of-a-zero problem.<ref name="ow"/> On the other hand, the exact pattern which repeats after <math>n > M</math> ''is'' computable.<ref name="ow"/><ref>{{Cite journal|last1=Berstel|first1=Jean|last2=Mignotte|first2=Maurice|date=1976|title=Deux propriétés décidables des suites récurrentes linéaires|journal=Bulletin de la Société Mathématique de France|language=fr|volume=104|pages=175–184|doi=10.24033/bsmf.1823|doi-access=free}}</ref> This is why the infinitely-many-zeros problem is decidable: just determine if the infinitely-repeating pattern is empty.
+Decidability results are known when the order of a sequence is restricted to be small. For example, the Skolem problem is decidable for sequences of order up to 4,<ref name="ow" /> and for a restricted set of sequences up to order 7.<ref name=LICS22/>
 ==Generalizations==
-A natural generalization is obtained by relaxing the condition that the coefficients of the recurrence are constants.  If the coefficients are allowed to be polynomials, then one obtains [[holonomic function|holonomic sequence]]s.
+A '''[[holonomic function|D-finite or holonomic sequence]]''' is a natural generalization where the coefficients of the recurrence are allowed to be polynomial functions of <math>n</math> rather than constants.<ref>{{Cite journal |last=Stanley |first=Richard P |date=1980 |title=Differentiably finite power series |journal=European journal of combinatorics |volume=1 |issue=2 |pages=175–188}}</ref>
-A [[k-regular sequence|<math>k</math>-regular sequence]] satisfies linear recurrences with constant coefficients, but the recurrences take a different form.  Rather than <math>s(n)</math> being a linear combination of <math>s(m)</math> for some integers <math>m</math> that are close to <math>n</math>, each term <math>s(n)</math> in a <math>k</math>-regular sequence is a linear combination of <math>s(m)</math> for some integers <math>m</math> whose base-<math>k</math> representations are close to that of <math>n</math>.  Constant-recursive sequences can be thought of as <math>1</math>-regular sequences, where the [[unary numeral system|base-1 representation]] of <math>n</math> consists of <math>n</math> copies of the digit <math>1</math>.
+A '''[[k-regular sequence|<math>k</math>-regular sequence]]''' satisfies a linear recurrences with constant coefficients, but the recurrences take a different form.  Rather than <math>s_n</math> being a linear combination of <math>s_m</math> for some integers <math>m</math> that are close to <math>n</math>, each term <math>s_n</math> in a <math>k</math>-regular sequence is a linear combination of <math>s_m</math> for some integers <math>m</math> whose [[radix|base]]-<math>k</math> representations are close to that of <math>n</math>.<ref>{{Cite journal |last=Allouche |first=Jean-Paul |last2=Shallit |first2=Jeffrey |date=1992 |title=The ring of k-regular sequences |journal=Theoretical Computer Science |volume=98 |issue=2 |pages=163–197}}</ref> Constant-recursive sequences can be thought of as <math>1</math>-regular sequences, where the [[unary numeral system|base-1 representation]] of <math>n</math> consists of <math>n</math> copies of the digit <math>1</math>.
-== Notes ==
+{{cn|date=April 2024}}
+==Notes==
 {{reflist}}
-== References ==
+==References==
+{{refbegin}}
-* {{cite book| first1=Alfred | last1=Brousseau
-|title=Linear Recursion and Fibonacci Sequences
+* {{cite book |last=Brousseau |first=Alfred |title=Linear Recursion and Fibonacci Sequences |year=1971 |publisher=Fibonacci Association |url=https://www.fq.math.ca/linear.html }}
+* {{Cite book |last1=Kauers |first1=Manuel |last2=Paule |first2=Peter |title=The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates |url=https://books.google.com/books?id=BPeODAEACAAJ |date=2010 |publisher=Springer Vienna |isbn=978-3-7091-0444-6 |language=en |pages=66 }}
-|year=1971
+* {{cite book |last=Stanley |first=Richard P. |date=2011 |title=Enumerative Combinatorics |url=https://www.ms.uky.edu/~sohum/putnam/enu_comb_stanley.pdf |volume=1 |edition=2 |publisher=Cambridge studies in advanced mathematics}}
-|publisher=Fibonacci Association
+{{refend}}
-|url=http://www.fq.math.ca/linear.html
-}}
-*{{cite book| first1=Ronald L. | last1=Graham |
-first2=Donald E. | last2=Knuth
-|first3=Oren | last3=Patashnik
-|title=Concrete Mathematics: A Foundation for Computer Science
-|edition=2
-|publisher=Addison-Wesley
-|year=1994 |isbn=978-0-201-55802-9
-| title-link=Concrete Mathematics }}
-== See also ==
-* [[Companion matrix]]
-== External links ==
-* {{cite web |title= OEIS Index Rec|url=http://oeis.org/wiki/Index_to_OEIS:_Section_Rec}} [[OEIS]] index to a few thousand examples of linear recurrences, sorted by order (number of terms) and signature (vector of values of the constant coefficients)
+[[Category:Combinatorics]]
+[[Category:Dynamical systems]]
 [[Category:Integer sequences]]
+[[Category:Linear algebra]]
 [[Category:Recurrence relations]]
+==External links==
+* {{cite web |title= OEIS Index Rec|url=http://oeis.org/wiki/Index_to_OEIS:_Section_Rec}} [[On-Line Encyclopedia of Integer Sequences|OEIS]] index to a few thousand examples of linear recurrences, sorted by order (number of terms) and signature (vector of values of the constant coefficients)