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A143008
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Crystal ball sequence for the A2 x A2 lattice.
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4
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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.
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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.
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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) [From Harvey P. Dale, June 14 2011]
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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.
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MAPLE
| p := n -> (3*n^4+6*n^3+9*n^2+6*n+2)/2: seq(p(n), n = 0..24);
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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] (* From Harvey P. Dale, June 14 2011 *)
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CROSSREFS
| Cf. A143007, A143009, A143010, A143011.
Sequence in context: A125258 A060886 A081586 * A107963 A006230 A066110
Adjacent sequences: A143005 A143006 A143007 * A143009 A143010 A143011
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KEYWORD
| easy,nonn
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AUTHOR
| Peter Bala (pbala(AT)toucansurf.com), Jul 22 2008
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EXTENSIONS
| More terms from Harvey P. Dale, June 14 2011.
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