The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A143010 Crystal ball sequence for the A4 x A4 lattice. 4
 1, 41, 661, 5741, 33001, 142001, 494341, 1465661, 3833941, 9073501, 19789001, 40328641, 77620661, 142282141, 250054001, 423621001, 694880441, 1107728161, 1721435341, 2614694501, 3890418001, 5681377241, 8156775661, 11529853541 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The A_4 lattice consists of all vectors v = (a,b,c,d,e) in Z^5 such that a+b+c+d+e = 0. The lattice is equipped with the norm ||v|| = 1/2*(|a| + |b| + |c| + |d| + |e|). Pairs of lattice points (v,w) in the product lattice A_4 x A_4 have norm ||(v,w)|| = ||v|| + ||w||. Then the k-th term in the crystal ball sequence for the A_4 x A_4 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, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142. Index entries for linear recurrences with constant coefficients, signature (9,-36, 84,-126,126,-84,36,-9,1). FORMULA a(n) = (35*n^8 +140*n^7 +630*n^6 +1400*n^5 +2595*n^4 +3020*n^3 +2500*n^2 +1200*n +288)/288 = 5*n*(n + 1)*(n^2 + n + 2)*(7*n^4 + 14*n^3 + 77*n^2 + 70*n + 120)/288 + 1. O.g.f. : 1/(1-x)*[Legendre_P(4,(1+x)/(1-x))]^2. Apery's constant zeta(3) = (1+1/2^3+1/3^3+1/4^3) + Sum {n = 1..inf} 1/(n^3*a(n-1)*a(n)). G.f.: (1+16*x+36*x^2+16*x^3+x^4)^2/(1-x)^9. [Colin Barker, Mar 16 2012] a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>8. - Vincenzo Librandi, Dec 16 2015 MAPLE p := n -> (35*n^8 +140*n^7 +630*n^6 +1400*n^5 +2595*n^4 +3020*n^3 +2500*n^2 +1200*n +288)/288: seq(p(n), n = 0..24); MATHEMATICA LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {1, 41, 661, 5741, 33001, 142001, 494341, 1465661, 3833941}, 25] (* Vincenzo Librandi, Dec 16 2015 *) PROG (Python) A143010_list, m = [], [4900, -14700, 17500, -10500, 3340, -540, 40, 0, 1] for _ in range(10**2):     A143010_list.append(m[-1])     for i in range(8):         m[i+1] += m[i] # Chai Wah Wu, Dec 15 2015 (MAGMA) [5*n*(n+1)*(n^2+n+2)*(7*n^4+14*n^3+77*n^2+70*n+120)/288+1: n in [0..30]]; // Vincenzo Librandi, Dec 16 2015 CROSSREFS Cf. A143007 (row 4), A143008, A143009, A143011. Sequence in context: A299600 A197371 A268748 * A009730 A009761 A118448 Adjacent sequences:  A143007 A143008 A143009 * A143011 A143012 A143013 KEYWORD easy,nonn AUTHOR Peter Bala, Jul 22 2008 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 January 21 01:45 EST 2022. Contains 350473 sequences. (Running on oeis4.)