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%I #9 May 28 2023 10:41:59
%S 1,25,253,1445,5741,17861,46705,107353,223465,430081,776821,1331485,
%T 2184053,3451085,5280521,7856881,11406865,16205353,22581805,30927061,
%U 41700541,55437845,72758753,94375625,121102201
%N Crystal ball sequence for the A3 x A3 lattice.
%C The A_3 lattice consists of all vectors v = (a,b,c,d) in Z^4 such that a+b+c+d = 0. The lattice is equipped with the norm ||v|| = 1/2*(|a| + |b| + |c| + |d|). Pairs of lattice points (v,w) in the product lattice A_3 x A_3 have norm ||(v,w)|| = ||v|| + ||w||. Then the k-th term in the crystal ball sequence for the A_3 x A_3 lattice gives the number of such pairs (v,w) for which ||(v,w)|| is less than or equal to k.
%H R. Bacher, P. de la Harpe and B. Venkov, <a href="http://archive.numdam.org/ARCHIVE/AIF/AIF_1999__49_3/AIF_1999__49_3_727_0/AIF_1999__49_3_727_0.pdf">Series de croissance et series d'Ehrhart associees aux reseaux de racines</a>, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7, -21, 35, -35, 21, -7, 1).
%F Row 3 of A143007. a(n) := (10*n^6+30*n^5+85*n^4+120*n^3+121*n^2+66*n+18)/18. O.g.f. : 1/(1-x)*[Legendre_P(3,(1+x)/(1-x))]^2. Apery's constant zeta(3) = (1+1/2^3+1/3^3) + sum {n = 1..inf} 1/(n^3*a(n-1)*a(n)).
%F G.f.: (1+x)^2*(1+8*x+x^2)^2/(1-x)^7. [Colin Barker, Mar 16 2012]
%p p := n -> (10*n^6+30*n^5+85*n^4+120*n^3+121*n^2+66*n+18)/18: seq(p(n), n = 0..24);
%Y Cf. A143007, A143008, A143010, A143011.
%K easy,nonn
%O 0,2
%A _Peter Bala_, Jul 22 2008
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