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 A143009 Crystal ball sequence for the A3 x A3 lattice. 4
 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 (list; graph; refs; listen; history; text; internal format)
 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. 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

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Last modified August 7 12:30 EDT 2022. Contains 355986 sequences. (Running on oeis4.)