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

Offset

0,2

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

Table of n, a(n) for n=0..24.

R. Bacher, P. de la Harpe and B. Venkov, Series de croissance et series d'Ehrhart associees aux reseaux 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).

Formula

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)).

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

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);

Crossrefs

Cf. A143007, A143008, A143010, A143011.

Sequence in context: A022685 A308492 A042208 * A090022 A298068 A017450

Adjacent sequences: A143006 A143007 A143008 * A143010 A143011 A143012

Keyword

easy,nonn

Author

Peter Bala, Jul 22 2008

Status

approved