Centered pentagonal number

{{Short description|Centered figurate number that represents a pentagon with a dot in the center}} {{refimprove|date=January 2025}} {{Use American English|date=March 2021}} {{Use mdy dates|date=March 2021}} 240px|right

In mathematics, 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.<ref>{{cite book |last=Weisstein |first=Eric W. |date=2002 |title= CRC Concise Encyclopedia of Mathematics |url=https://www.google.com/books/edition/CRC_Concise_Encyclopedia_of_Mathematics/D_XKBQAAQBAJ |publisher=CRC Press |page=367 |isbn=9781420035223 |access-date=January 25, 2025}}</ref> The centered pentagonal number for ''n'' is given by the formula

:<math>P_{n}={{5n^2 + 5n + 2} \over 2}, n\geq1</math>

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 {{OEIS|A005891}}.

==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. *Centered pentagonal numbers follow the following recurrence relations: :<math>P_{n}=P_{n-1}+5n , P_0=1</math> :<math>P_{n}=3(P_{n-1}-P_{n-2})+P_{n-3} , P_0=1,P_1=6,P_2=16</math> *Centered pentagonal numbers can be expressed using triangular numbers: :<math>P_{n}=5T_n+1</math>

==References== {{reflist}}

==External links== *{{mathworld|title=Centered pentagonal number|urlname=CenteredPentagonalNumber}}

Category:Figurate numbers