A365204 -id:A365204 - cp's OEIS frontend

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

Léo Cymrot Cymbalista, Aug 25 2023

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.

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).

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

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

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

A365206 Centered octachoral numbers.

Original entry on oeis.org

1, 49, 337, 1249, 3361, 7441, 14449, 25537, 42049, 65521, 97681, 140449, 195937, 266449, 354481, 462721, 594049, 751537, 938449, 1158241, 1414561, 1711249, 2052337, 2442049, 2884801, 3385201, 3948049, 4578337, 5281249, 6062161, 6926641, 7880449, 8929537

Offset: 1

Léo Cymrot Cymbalista, Aug 25 2023

A octachoral number is a centered figurate number that represents a octachoron, which is a four-dimensional regular polytope composed of 8 cells (also known as tesseract).

Eric Weisstein's World of Mathematics, Figurate Number.
Wikipedia, 8-cell.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001844 (2D), A005917 (3D), A362863, A365204, A365205.

Table[8*n^4 - 16*n^3 + 16*n^2 - 8n + 1, {n, 1, 100}]

a(n) = 8*n^4 - 16*n^3 + 16*n^2 - 8n + 1.

G.f.: x*(1 + 44*x + 102*x^2 + 44*x^3 + x^4)/(1 - x)^5. - Stefano Spezia, Aug 26 2023

cp's OEIS Frontend

A365205 Centered pentachoral numbers.

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