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

Offset: 1

Author

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

Keywords

• easy
• nonn

Comments

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

Examples

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

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). [_R. J. Mathar_, Jun 21 2010]

Crossrefs

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

Programs

Magma

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

Mathematica

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

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)

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. [R. J. Mathar, Jun 21 2010]