Pyramidal number: Difference between revisions

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
VisualWikitext
imported>David Eppstein
m Reverted edits by Radlrb (talk) to last version by Imaginatorium
 
imported>Decent Direction
 
Line 7: Line 7:


:<math>P_n^r= \frac{3n^2 + n^3(r-2) - n(r-5)}{6},</math>
:<math>P_n^r= \frac{3n^2 + n^3(r-2) - n(r-5)}{6},</math>
where {{math|''r'' ∈ <math>\mathbb{N}</math>}}, {{math|''r'' ≥ 3}}. <ref name=":0">{{MathWorld |id=PyramidalNumber |title=Pyramidal Number}}</ref>
where <math>r \isin \mathbb{N}</math>, {{math|''r'' ≥ 3}}.<ref name=":0">{{MathWorld |id=PyramidalNumber |title=Pyramidal Number}}</ref>


This formula can be factored:
This formula can be factored:
Line 25: Line 25:
The first few pentagonal pyramidal numbers are:
The first few pentagonal pyramidal numbers are:


:[[1 (number)|1]], [[6 (number)|6]], [[18 (number)|18]], [[40 (number)|40]], [[75 (number)|75]], [[126 (number)|126]], [[196 (number)|196]], [[288 (number)|288]], 405, 550, 726, 936, 1183, 1470, 1800, 2176, 2601, 3078, 3610, 4200, 4851, 5566, 6348, 7200, 8125, 9126 {{OEIS|id=A002411}}.
:[[1 (number)|1]], [[6 (number)|6]], [[18 (number)|18]], [[40 (number)|40]], [[75 (number)|75]], [[126 (number)|126]], [[196 (number)|196]], [[288 (number)|288]], 405, 550, 726, 936, 1183, ... {{OEIS|id=A002411}}.


The first few hexagonal pyramidal numbers are:
The first few hexagonal pyramidal numbers are:

Latest revision as of 02:22, 1 August 2025

Template:Short description

File:Square pyramidal number.svg
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

The formula for the Template:Mvarth Template:Mvar-gonal pyramidal number is

Pnr=3n2+n3(r−2)−n(r−5)6,

where r∈ℕ, r ≥ 3Script error: No such module "Check for unknown parameters"..[1]

This formula can be factored:

Pnr=n(n+1)(n(r−2)−(r−5))(2)(3)=(n(n+1)2)(n(r−2)−(r−5)3)=Tn⋅n(r−2)−(r−5)3,

where Template:Mvar is the Template:Mvarth triangular number.

Sequences

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, ... (sequence A002411 in the OEIS).

The first few hexagonal pyramidal numbers are:

Template:Num, Template:Num, Template:Num, Template:Num, Template:Num, Template:Num, Template:Num, 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

<templatestyles src="Reflist/styles.css" />

  1. ↑ a b Script error: No such module "Template wrapper".
  2. ↑ Script error: No such module "citation/CS1".Script error: No such module "Check for unknown parameters".
  3. ↑ Script error: No such module "citation/CS1"..

Script error: No such module "Check for unknown parameters".

Template:Figurate numbers

Retrieved from "http://debianws.lexgopc.com/wiki143/index.php?title=Pyramidal_number&oldid=4955580"