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A143009 Crystal ball sequence for the A3 x A3 lattice.

%I A143009 #23 Aug 21 2025 16:50:27
%S A143009 1,25,253,1445,5741,17861,46705,107353,223465,430081,776821,1331485,
%T A143009 2184053,3451085,5280521,7856881,11406865,16205353,22581805,30927061,
%U A143009 41700541,55437845,72758753,94375625,121102201,153862801,193701925,241794253,299455045,368150941,449511161,545339105,657624353,788555065,940530781,1116175621,1318351885,1550174053,1815023185,2116561721
%N A143009 Crystal ball sequence for the A3 x A3 lattice.
%C A143009 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 A143009 Paolo Xausa, <a href="/A143009/b143009.txt">Table of n, a(n) for n = 0..10000</a>
%H A143009 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">Séries de croissance et séries d'Ehrhart associées aux réseaux de racines</a>, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
%H A143009 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A143009 a(n) = (10*n^6+30*n^5+85*n^4+120*n^3+121*n^2+66*n+18)/18.
%F A143009 O.g.f.: 1/(1-x)*[Legendre_P(3,(1+x)/(1-x))]^2.
%F A143009 Apery's constant zeta(3) = (1+1/2^3+1/3^3) + Sum_{n >= 1} 1/(n^3*a(n-1)*a(n)).
%F A143009 G.f.: (1+x)^2*(1+8*x+x^2)^2/(1-x)^7. - _Colin Barker_, Mar 16 2012
%p A143009 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);
%t A143009 A143009[n_] := n*(n + 1)*(5*n*(n + 1)*(2*n*(n + 1) + 11) + 66)/18 + 1;
%t A143009 Array[A143009, 50, 0] (* _Paolo Xausa_, Aug 21 2025 *)
%Y A143009 Row 3 of A143007.
%Y A143009 Cf. A143008, A143010, A143011.
%K A143009 easy,nonn
%O A143009 0,2
%A A143009 _Peter Bala_, Jul 22 2008