%I #17 May 04 2024 00:29:32
%S 1,120,947,3652,9985,22276,43435,76952,126897,197920,295251,424700,
%T 592657,806092,1072555,1400176,1797665,2274312,2839987,3505140,
%U 4280801,5178580,6210667,7389832,8729425,10243376,11946195,13852972,15979377
%N Figurate numbers based on the 600-cell (4-D polytope with Schlaefli symbol {3,3,5}).
%C This is the 4-dimensional regular convex polytope called the 600-cell, hexacosichoron or hypericosahedron.
%H Vincenzo Librandi, <a href="/A092182/b092182.txt">Table of n, a(n) for n = 1..1000</a>
%H Hyun Kwang Kim, <a href="http://dx.doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer. Math. Soc., 131 (2002), 65-75.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/600-Cell.html">600-Cell</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1). [_R. J. Mathar_, Jun 21 2010]
%F a(n) = n*((145*n^3)-(280*n^2)+(179*n)-38)/6
%F a(n) = C(n+3,4) + 115 C(n+2,4) + 357 C(n+1,4) + 107 C(n,4)
%F 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]
%e a(3)= 3*((145*3^3)-(280*3^2)+(179*3)-38)/6 = 3*(3915-2520+537-38)/6 = 0.5*1894 = 947
%t LinearRecurrence[{5,-10,10,-5,1},{1,120,947,3652,9985},30] (* _Harvey P. Dale_, May 04 2024 *)
%o (Magma) [n*((145*n^3)-(280*n^2)+(179*n)-38)/6: n in [1..40]]; // _Vincenzo Librandi_, May 22 2011
%Y Cf. A000332, A000583, A014820, A092181, A092183.
%K easy,nonn
%O 1,2
%A Michael J. Welch (mjw1(AT)ntlworld.com), Mar 31 2004