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A092183
Figurate numbers based on the 120-cell (4-D polytope with Schlaefli symbol {5,3,3}).
8
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, 64495575, 74791108, 86272593, 99027780
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
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
Eric Weisstein's World of Mathematics, 120-Cell.
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