A092183 Figurate numbers based on the 120-cell (4-D polytope with Schlaefli symbol {5,3,3}).

1, 600, 4983, 19468, 53505, 119676, 233695, 414408, 683793, 1066960, 1592151, 2290740, 3197233, 4349268, 5787615, 7556176, 9701985, 12275208, 15329143, 18920220, 23108001, 27955180, 33527583, 39894168, 47127025, 55301376

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 120-cell, hecatonicosachoron or hyperdodecahedron.

Examples

			a(3) = 3*((261*3^3)-(504*3^2)+(283*3)-38)/2 = 3*(7047-4536+849-38)/2 = 1.5*3322 = 4983
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
• Eric Weisstein's World of Mathematics, 120-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, A092182.

Programs

Magma

[n*((261*n^3)-(504*n^2)+(283*n)-38)/2: n in [1..40]]; // Vincenzo Librandi, May 22 2011

Mathematica

Table[SeriesCoefficient[x (1 + 595 x + 1993 x^2 + 543 x^3)/(1 - x)^5, {x, 0, n}], {n, 26}] (* Michael De Vlieger, Dec 14 2015 *)

PARI

a(n) = n*(261*n^3 - 504*n^2 + 283*n - 38)/2; \\ Michel Marcus, Dec 14 2015

Formula

a(n) = n*((261*n^3)-(504*n^2)+(283*n)-38)/2.

a(n) = C(n+3,4) + 595 C(n+2,4) + 1993 C(n+1,4) + 543 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+595*x+1993*x^2+543*x^3)/(1-x)^5. [R. J. Mathar, Jun 21 2010]