A143008 Crystal ball sequence for the A2 x A2 lattice.

1, 13, 73, 253, 661, 1441, 2773, 4873, 7993, 12421, 18481, 26533, 36973, 50233, 66781, 87121, 111793, 141373, 176473, 217741, 265861, 321553, 385573, 458713, 541801, 635701, 741313, 859573, 991453, 1137961, 1300141, 1479073, 1675873, 1891693, 2127721

Offset: 0

Author

Peter Bala, Jul 22 2008

Keywords

• easy
• nonn

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.

Examples

			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.
		

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).

Crossrefs

Cf. A143007, A143009, A143010, A143011.

Programs

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 *)

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

Extensions

More terms from Harvey P. Dale, Jun 14 2011