A092182 Figurate numbers based on the 600-cell (4-D polytope with Schlaefli symbol {3,3,5}).

1, 120, 947, 3652, 9985, 22276, 43435, 76952, 126897, 197920, 295251, 424700, 592657, 806092, 1072555, 1400176, 1797665, 2274312, 2839987, 3505140, 4280801, 5178580, 6210667, 7389832, 8729425, 10243376, 11946195, 13852972, 15979377, 18341660, 20956651, 23841760

Offset

1,2

Comments

This is the 4-dimensional regular convex polytope called the 600-cell, hexacosichoron or hypericosahedron.

References

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 191.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.

Eric Weisstein's World of Mathematics, 600-Cell

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = n*((145*n^3)-(280*n^2)+(179*n)-38)/6.

a(n) = C(n+3,4) + 115 C(n+2,4) + 357 C(n+1,4) + 107 C(n,4).

From R. J. Mathar, Jun 21 2010: (Start)

a(n) = +5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4) +a(n-5).

G.f.: x*(1+115*x+357*x^2+107*x^3)/(1-x)^5. (End)

E.g.f.: exp(x)*x*(6 + 354*x + 590*x^2 + 145*x^3)/6. - Stefano Spezia, Oct 26 2025

Example

a(3) = 3*((145*3^3)-(280*3^2)+(179*3)-38)/6 = 3*(3915-2520+537-38)/6 = 0.5*1894 = 947.

Mathematica

LinearRecurrence[{5, -10, 10, -5, 1}, {1, 120, 947, 3652, 9985}, 30] (* Harvey P. Dale, May 04 2024 *)

Prog

(Magma) [n*((145*n^3)-(280*n^2)+(179*n)-38)/6: n in [1..40]]; // Vincenzo Librandi, May 22 2011

Crossrefs

Cf. A000332, A000583, A014820, A092181, A092183.

Sequence in context: A250512 A011245 A213875 * A262387 A133119 A052777

Adjacent sequences: A092179 A092180 A092181 * A092183 A092184 A092185

Keyword

easy,nonn

Author

Michael J. Welch (mjw1(AT)ntlworld.com), Mar 31 2004

Status

approved