Centered pentagonal number

{{Short description|Centered figurate number that represents a pentagon with a dot in the center}}

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File:Nombre pentagon cent.svg

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.{{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}} The centered pentagonal number for n is given by the formula

:$P\_\{n\}=\{\{5n^2\; -\; 5n\; +\; 2\}\; \backslash over\; 2\},\; n\backslash geq1$

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}}.