A365205 Centered pentachoral numbers.

1, 21, 121, 421, 1101, 2401, 4621, 8121, 13321, 20701, 30801, 44221, 61621, 83721, 111301, 145201, 186321, 235621, 294121, 362901, 443101, 535921, 642621, 764521, 903001, 1059501, 1235521, 1432621, 1652421, 1896601, 2166901, 2465121, 2793121, 3152821

Offset: 1

Author

Léo Cymrot Cymbalista, Aug 25 2023

Keywords

• nonn
• easy

Comments

A pentachoral number is a centered figurate number that represents a pentachoron, which is a four-dimensional regular polytope composed of 5 cells.

One of the 6 centered regular polichoral (centered pentachoral, centered hexadecachoral, centered octachoral, centered icositetrachoral, centered hexacosichoral and centered hecatonicosachoral) numbers.

Links

• Eric Weisstein's World of Mathematics, Figurate Number.
• Wikipedia, 5-cell.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A005448 (2D), A005898 (3D), A362863, A365204, A365206.

Programs

Mathematica

Table[5/2*n^4 - 5*n^3 + 15/2*n^2 - 5*n + 1, {n, 1, 100}]

Formula

a(n) = 5*n^4/2 - 5*n^3 + 15*n^2/2 - 5*n + 1.

G.f.: x*(1 + 16*x + 26*x^2 + 16*x^3 + x^4)/(1 - x)^5. - Stefano Spezia, Aug 26 2023