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
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
KEYWORD
easy,nonn
AUTHOR
Peter Bala, Jul 22 2008
STATUS
approved