pyramidal number

File:Square pyramidal number.svgA pyramidal number is the number of points in a pyramid with a polygonal base and triangular sides. 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.{{Cite OEIS|1=A002414}} 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

The formula for the {{mvar|n}}th {{mvar|r}}-gonal pyramidal number is

: $P_n^r= \frac{3n^2 + n^3(r-2) - n(r-5)}{6},$

where {{math|r ∈ $\mathbb{N}$ }}, {{math|r ≥ 3}}. {{MathWorld |id=PyramidalNumber |title=Pyramidal Number}}

This formula can be factored:

: $P_n^r=\frac{n(n+1)\bigl(n(r-2)-(r-5)\bigr)}{(2)(3)}=\left(\frac{n(n+1)}{2}\right)\left(\frac{n(r-2)-(r-5)}{3}\right)=T_n \cdot \frac{n(r-2)-(r-5)}{3},$

where {{mvar|T_n}} is the {{mvar|n}}th triangular number.

Sequences

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

:1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ... {{OEIS|id=A000292}}

The first few square pyramidal numbers are:

:1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, ... {{OEIS|id=A000330}}.

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 {{OEIS|id=A002411}}.

The first few hexagonal pyramidal numbers are:

:{{num|1}}, {{num|7}}, {{num|22}}, {{num|50}}, {{num|95}}, {{num|161}}, {{num|252}}, 372, 525, 715, 946, 1222, 1547, 1925 {{OEIS|A002412}}.

The first few heptagonal pyramidal numbers are:{{citation|title=Recreations in the Theory of Numbers: The Queen of Mathematics Entertains|first=Albert H.|last=Beiler|publisher=Courier Dover Publications|year=1966|isbn=9780486210964|page=194|url=https://books.google.com/books?id=fJTifbYNOzUC&pg=PA194}}.

:1, 8, 26, 60, 115, 196, 308, 456, 645, 880, 1166, 1508, 1911, ... {{OEIS|id=A002413}}

References

Category:Figurate numbers