OFFSET
0,2
COMMENTS
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.
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 (5, -10, 10, -5, 1).
FORMULA
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)).
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
G.f.: (1+4*x+x^2)^2/(1-x)^5. - Colin Barker, Feb 22 2012
EXAMPLE
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.
MAPLE
p := n -> (3*n^4+6*n^3+9*n^2+6*n+2)/2: seq(p(n), n = 0..24);
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Peter Bala, Jul 22 2008
EXTENSIONS
More terms from Harvey P. Dale, Jun 14 2011
STATUS
approved