Pyramidal number

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title: "Pyramidal number" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["figurate-numbers"] description: "Figurate number" topic_path: "general/figurate-numbers" source: "https://en.wikipedia.org/wiki/Pyramidal_number" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Figurate number ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/4/47/Square_pyramidal_number.svg" caption="1=A002414}} The numbers of points in the base and in layers parallel to the base are given by [[polygonal number]]s 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 nth r-gonal pyramidal number is

:P_n^r= \frac{3n^2 + n^3(r-2) - n(r-5)}{6}, where r \isin \mathbb{N}, r ≥ 3.

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 Tn is the nth triangular number.

Sequences

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

:1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ...

The first few square pyramidal numbers are: :1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, ... .

The first few pentagonal pyramidal numbers are:

:1, 6, 18, 40, 75, 126, 196, 288, 405, 550, 726, 936, 1183, ... .

The first few hexagonal pyramidal numbers are: :, , , , , , , 372, 525, 715, 946, 1222, 1547, 1925 .

The first few heptagonal pyramidal numbers are: :1, 8, 26, 60, 115, 196, 308, 456, 645, 880, 1166, 1508, 1911, ...

References

References

1. {{Cite OEIS
2. "Pyramidal Number".
3. Beiler, Albert H.. (1966). "Recreations in the Theory of Numbers: The Queen of Mathematics Entertains". Courier Dover Publications.

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::