A362863 Centered hecatonicosachoral numbers.

1, 1441, 11521, 44641, 122401, 273601, 534241, 947521, 1563841, 2440801, 3643201, 5243041, 7319521, 9959041, 13255201, 17308801, 22227841, 28127521, 35130241, 43365601, 52970401, 64088641, 76871521, 91477441, 108072001, 126828001, 147925441, 171551521, 197900641

Offset: 1

Author

Léo Cymrot Cymbalista, May 06 2023

Keywords

• nonn
• easy

Comments

A hecatonicosachoral number is a centered figurate number that represents a hecatonicosachoron, which is a four-dimensional regular polytope composed of 120 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, 120-cell.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A005891 (2D), A005904 (3D), A006322, A151989.

Programs

Mathematica

Table[300*n^4 - 600*n^3 + 420*n^2 - 120*n + 1, {n, 1, 100}]

Formula

a(n) = 300*n^4 - 600*n^3 + 420*n^2 - 120*n + 1.

a(n) = 1440*A006322(n-1) + 1 for n > 1.

a(n) = 288*(A151989(n-1)-1)/25 + 1.

G.f.: x*(1 + 1436*x + 4326*x^2 + 1436*x^3 + x^4)/(1 - x)^5. - Stefano Spezia, May 12 2023