Editing 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}}
[[File:Nombre pentagon cent.svg|240px|right]]

In [[mathematics]], a '''centered pentagonal number''' is a [[centered polygonal number|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 (number)|1]], [[6 (number)|6]], [[16 (number)|16]], [[31 (number)|31]], [[51 (number)|51]], [[76 (number)|76]],
[[106 (number)|106]], [[141 (number)|141]], [[181 (number)|181]], [[226 (number)|226]], [[276 (number)|276]],
[[331 (number)|331]], [[391 (number)|391]], [[456 (number)|456]], [[526 (number)|526]], [[601 (number)|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 Number|triangular numbers]]:
:<math>P_{n}=5T_{n-1}+1</math>

==References==
{{reflist}}

==See also==
*[[Pentagonal number]]
*[[Polygonal number]]
*[[Centered polygonal number]]

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

[[Category:Figurate numbers]]

{{Figurate numbers}}
{{Classes of natural numbers}}

{{num-stub}}