A143009 Crystal ball sequence for the A3 x A3 lattice.

1, 25, 253, 1445, 5741, 17861, 46705, 107353, 223465, 430081, 776821, 1331485, 2184053, 3451085, 5280521, 7856881, 11406865, 16205353, 22581805, 30927061, 41700541, 55437845, 72758753, 94375625, 121102201, 153862801, 193701925, 241794253, 299455045, 368150941, 449511161, 545339105, 657624353, 788555065, 940530781, 1116175621, 1318351885, 1550174053, 1815023185, 2116561721

Offset: 0

Author

Peter Bala, Jul 22 2008

Keywords

• easy
• nonn
• changed

Comments

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.

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000
• R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Row 3 of A143007.

Cf. A143008, A143010, A143011.

Programs

Maple

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);

Mathematica

A143009[n_] := n*(n + 1)*(5*n*(n + 1)*(2*n*(n + 1) + 11) + 66)/18 + 1;
Array[A143009, 50, 0] (* Paolo Xausa, Aug 21 2025 *)

Formula

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} 1/(n^3*a(n-1)*a(n)).

G.f.: (1+x)^2*(1+8*x+x^2)^2/(1-x)^7. - Colin Barker, Mar 16 2012