OFFSET
0,3
COMMENTS
This is the 4-dimensional regular convex polytope called the 24-cell, hyperdiamond or icositetrachoron.
REFERENCES
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 193.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
Eric Weisstein's World of Mathematics, 24-Cell.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
a(n) = n^2*(3*n^2 - 4*n + 2).
a(n) = C(n+3,4) + 19*C(n+2,4) + 43*C(n+1,4) + 9*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+19*x+43*x^2+9*x^3)/(1-x)^5. (End)
a(n) = Sum_{k = 1..n} (k^3 + k^7)* binomial(n,k)/binomial(n+k,k). Cf. A034262 and A155977. - Peter Bala, Feb 12 2019
E.g.f.: exp(x)*x*(1 + 11*x + 14*x^2 + 3*x^3). - Stefano Spezia, Oct 27 2025
EXAMPLE
a(3) = 3^2*((3*3^2)-(4*3)+2) = 9*(27-12+2) = 9*17 = 153
MATHEMATICA
Table[SeriesCoefficient[x (1 + 19 x + 43 x^2 + 9 x^3)/(1 - x)^5, {x, 0, n}], {n, 0, 32}] (* Michael De Vlieger, Dec 14 2015 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 24, 153, 544}, 40] (* Harvey P. Dale, May 25 2022 *)
PROG
(Magma) [n^2*((3*n^2)-(4*n)+2): n in [0..40]]; // Vincenzo Librandi, May 22 2011
(PARI) a(n) = n^2*(3*n^2-4*n+2); \\ Michel Marcus, Dec 14 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Michael J. Welch (mjw1(AT)ntlworld.com), Mar 31 2004
EXTENSIONS
Initial term 0 prepended by Kelvin Voskuijl, Sep 29 2025
STATUS
approved