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%I #16 Jul 31 2015 21:38:00
%S 1,13,73,253,661,1441,2773,4873,7993,12421,18481,26533,36973,50233,
%T 66781,87121,111793,141373,176473,217741,265861,321553,385573,458713,
%U 541801,635701,741313,859573,991453,1137961,1300141,1479073,1675873,1891693,2127721
%N Crystal ball sequence for the A2 x A2 lattice.
%C The A_2 lattice consists of all vectors v = (a,b,c) in Z^3 such that a+b+c = 0. The lattice is equipped with the norm ||v|| = 1/2*(|a| + |b| + |c|). Pairs of lattice points (v,w) in the product lattice A_2 x A_2 have norm ||(v,w)|| = ||v|| + ||w||. Then the k-th term in the crystal ball sequence for the A_2 x A_2 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_05">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5, 1).
%F Row 2 of A143007. a(n) := (3*n^4+6*n^3+9*n^2+6*n+2)/2. O.g.f. : 1/(1-x)*[Legendre_P(2,(1+x)/(1-x))]^2. Apery's constant zeta(3) = 9/8 + sum {n = 1..inf} 1/(n^3*a(n-1)*a(n)).
%F a(0)=1, a(1)=13, a(2)=73, a(3)=253, a(4)=661, a(n)=5*a(n-1)-10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5). - _Harvey P. Dale_, Jun 14 2011
%F G.f.: (1+4*x+x^2)^2/(1-x)^5. - _Colin Barker_, Feb 22 2012
%e a(1) = 13. a(1) gives the number of pairs of vectors (v,w) in the hyperplane a+b+c = 0 in Z^3 with ||v||+||w|| <= 1. Either v = w = (0,0,0), or v = (0,0,0) and w is one of the six possibilities (0,1,-1), (0,-1,1), (1,0,-1), (1,-1,0), (-1,0,1), (-1,1,0) or, alternatively, w =(0,0,0) and v equals one of these six possibilities.
%p p := n -> (3*n^4+6*n^3+9*n^2+6*n+2)/2: seq(p(n), n = 0..24);
%t Table[(3n^4+6n^3+9n^2+6n+2)/2,{n,0,45}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{1,13,73,253,661},45] (* _Harvey P. Dale_, Jun 14 2011 *)
%Y Cf. A143007, A143009, A143010, A143011.
%K easy,nonn
%O 0,2
%A _Peter Bala_, Jul 22 2008
%E More terms from _Harvey P. Dale_, Jun 14 2011
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