Cannabaceae

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Geometric representation of the square pyramidal number 1 + 4 + 9 + 16 = 30.

A pyramidal number is the number of points in a pyramid with a polygonal base and triangular sides.[1] The term often refers to square pyramidal numbers, which have a square base with four sides, but it can also refer to a pyramid with any number of sides.[2] The numbers of points in the base and in layers parallel to the base are given by polygonal numbers of the given number of sides, while the numbers of points in each triangular side is given by a triangular number. It is possible to extend the pyramidal numbers to higher dimensions.

Formula

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The formula for the nth r-gonal pyramidal number is

where r ∈ , r ≥ 3. [1]

This formula can be factored:

where Tn is the nth triangular number.

Sequences

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The first few triangular pyramidal numbers (equivalently, tetrahedral numbers) are:

1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ... (sequence A000292 in the OEIS)

The first few square pyramidal numbers are:

1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, ... (sequence A000330 in the OEIS).

The first few pentagonal pyramidal numbers are:

1, 6, 18, 40, 75, 126, 196, 288, 405, 550, 726, 936, 1183, 1470, 1800, 2176, 2601, 3078, 3610, 4200, 4851, 5566, 6348, 7200, 8125, 9126 (sequence A002411 in the OEIS).

The first few hexagonal pyramidal numbers are:

1, 7, 22, 50, 95, 161, 252, 372, 525, 715, 946, 1222, 1547, 1925 (sequence A002412 in the OEIS).

The first few heptagonal pyramidal numbers are:[3]

1, 8, 26, 60, 115, 196, 308, 456, 645, 880, 1166, 1508, 1911, ... (sequence A002413 in the OEIS)

References

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  1. ^ a b Weisstein, Eric W. "Pyramidal Number". MathWorld.
  2. ^ Sloane, N. J. A. (ed.). "Sequence A002414". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  3. ^ Beiler, Albert H. (1966), Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, Courier Dover Publications, p. 194, ISBN 9780486210964.
source: Https://en.wikipedia.org/wiki/Pyramidal_number

One thought on “Cannabaceae”

  1. Well, that’s interesting to know that Psilotum nudum are known as whisk ferns. Psilotum nudum is the commoner species of the two. While the P. flaccidum is a rare species and is found in the tropical islands. Both the species are usually epiphytic in habit and grow upon tree ferns. These species may also be terrestrial and grow in humus or in the crevices of the rocks.
    View the detailed Guide of Psilotum nudum: Detailed Study Of Psilotum Nudum (Whisk Fern), Classification, Anatomy, Reproduction

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