OFFSET
1,2
COMMENTS
This is the 4-dimensional regular convex polytope called the 120-cell, hecatonicosachoron or hyperdodecahedron.
REFERENCES
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 192.
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).
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).
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+595*x+1993*x^2+543*x^3)/(1-x)^5. (End)
E.g.f.: exp(x)*x*(2 + 598*x + 1062*x^2 + 261*x^3)/2. - Stefano Spezia, Oct 27 2025
EXAMPLE
a(3) = 3*((261*3^3)-(504*3^2)+(283*3)-38)/2 = 3*(7047-4536+849-38)/2 = 1.5*3322 = 4983
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 *)
PROG
(Magma) [n*((261*n^3)-(504*n^2)+(283*n)-38)/2: n in [1..40]]; // Vincenzo Librandi, May 22 2011
(PARI) a(n) = n*(261*n^3 - 504*n^2 + 283*n - 38)/2; \\ Michel Marcus, Dec 14 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Michael J. Welch (mjw1(AT)ntlworld.com), Mar 31 2004
STATUS
approved