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%I #22 Sep 08 2022 08:45:13
%S 1,24,153,544,1425,3096,5929,10368,16929,26200,38841,55584,77233,
%T 104664,138825,180736,231489,292248,364249,448800,547281,661144,
%U 791913,941184,1110625,1301976,1517049,1757728,2025969,2323800,2653321,3016704
%N Figurate numbers based on the 24-cell (4-D polytope with Schlaefli symbol {3,4,3}).
%C This is the 4-dimensional regular convex polytope called the 24-cell, hyperdiamond or icositetrachoron.
%H Vincenzo Librandi, <a href="/A092181/b092181.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 (2003), 65-75.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/24-Cell.html">24-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^2*((3*n^2)-(4*n)+2).
%F a(n) = C(n+3,4) + 19 C(n+2,4) + 43 C(n+1,4) + 9 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+19*x+43*x^2+9*x^3)/(1-x)^5. [_R. J. Mathar_, Jun 21 2010]
%F 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 a(3)= 3^2*((3*3^2)-(4*3)+2) = 9*(27-12+2) = 9*17 = 153
%t Table[SeriesCoefficient[x (1 + 19 x + 43 x^2 + 9 x^3)/(1 - x)^5, {x, 0, n}], {n, 32}] (* _Michael De Vlieger_, Dec 14 2015 *)
%t LinearRecurrence[{5,-10,10,-5,1},{1,24,153,544,1425},40] (* _Harvey P. Dale_, May 25 2022 *)
%o (Magma) [n^2*((3*n^2)-(4*n)+2): n in [1..40]]; // _Vincenzo Librandi_, May 22 2011
%o (PARI) a(n) = n^2*(3*n^2-4*n+2); \\ _Michel Marcus_, Dec 14 2015
%Y Cf. A000332, A000583, A014820, A092182, A092183.
%K easy,nonn
%O 1,2
%A Michael J. Welch (mjw1(AT)ntlworld.com), Mar 31 2004
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