Pyramidal number

Figurate number

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

P n r = 3 n 2 + n 3 ( r − 2 ) − n ( r − 5 ) 6 , {\displaystyle P_{n}^{r}={\frac {3n^{2}+n^{3}(r-2)-n(r-5)}{6}},}

where r ∈  N {\displaystyle \mathbb {N} } , r ≥ 3. [1]

This formula can be factored:

P n r = n ( n + 1 ) ( n ( r − 2 ) − ( r − 5 ) ) ( 2 ) ( 3 ) = ( n ( n + 1 ) 2 ) ( n ( r − 2 ) − ( r − 5 ) 3 ) = T n ⋅ n ( r − 2 ) − ( r − 5 ) 3 , {\displaystyle P_{n}^{r}={\frac {n(n+1){\bigl (}n(r-2)-(r-5){\bigr )}}{(2)(3)}}=\left({\frac {n(n+1)}{2}}\right)\left({\frac {n(r-2)-(r-5)}{3}}\right)=T_{n}\cdot {\frac {n(r-2)-(r-5)}{3}},}

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.

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Figurate numbers

2-dimensional|

| centered|

* Centered triangular numbers
* Centered square numbers
* Centered pentagonal numbers
* Centered hexagonal numbers
* Centered heptagonal numbers
* Centered octagonal numbers
* Centered nonagonal numbers
* Centered decagonal numbers
* Star numbers

|

non-centered|

* Triangular numbers
* Square numbers
* Pentagonal numbers
* Hexagonal numbers
* Heptagonal numbers
* Octagonal numbers
* Nonagonal numbers
* Decagonal numbers
* Dodecagonal numbers

3-dimensional|

| centered|

* Centered tetrahedral numbers
* Centered cube numbers
* Centered octahedral numbers
* Centered dodecahedral numbers
* Centered icosahedral numbers

|

non-centered|

* Cube numbers
* Octahedral numbers
* Dodecahedral numbers
* Icosahedral numbers
* Stella octangula numbers

pyramidal|

* Tetrahedral numbers
* Square pyramidal numbers

4-dimensional|

| non-centered|

* Pentatope numbers
* Squared triangular numbers
* Tesseractic numbers

|

Higher dimensional|

| non-centered|

* 5-hypercube numbers
* 6-hypercube numbers
* 7-hypercube numbers
* 8-hypercube numbers

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