This site is supported by donations to The OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A143008 Crystal ball sequence for the A2 x A2 lattice. 5
 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; text; internal format)
 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 Cf. A143007, A143009, A143010, A143011. Sequence in context: A125258 A060886 A081586 * A107963 A006230 A066110 Adjacent sequences:  A143005 A143006 A143007 * A143009 A143010 A143011 KEYWORD easy,nonn AUTHOR Peter Bala, Jul 22 2008 EXTENSIONS More terms from Harvey P. Dale, Jun 14 2011 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified March 25 08:15 EDT 2019. Contains 321469 sequences. (Running on oeis4.)