Centered pentagonal number

This article needs additional citations for verification. Please help by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources:  –  ·  ·  · scholar · JSTOR (August 2012) (Learn how and when to remove this template message)

A centered pentagonal number is a centered figurate number that represents a pentagon with a dot in the center and all other dots surrounding the center in successive pentagonal layers. The centered pentagonal number for n is given by the formula

${\displaystyle P_{n}={{5n^{2}-5n+2} \over 2},n\geq 1}$

The first few centered pentagonal numbers are

1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, 391, 456, 526, 601, 681, 766, 856, 951, 1051, 1156, 1266, 1381, 1501, 1626, 1756, 1891, 2031, 2176, 2326, 2481, 2641, 2806, 2976 (sequence A005891 in the OEIS).

Properties[]

• The parity of centered pentagonal numbers follows the pattern odd-even-even-odd, and in base 10 the units follow the pattern 1-6-6-1.
• pentagonal numbers follow the Recurrence relation:

${\displaystyle P_{n}=3(P_{n-1}-P_{n-2})+P_{n-3},P_{1}=1,P_{2}=6,P_{3}=16}$

See also[]

• Pentagonal number
• Polygonal number
• Centered polygonal number

External links[]

• Weisstein, Eric W. "Centered pentagonal number". MathWorld.

This article about a number is a stub. You can help Wikipedia by .

• v
• t