Search: id:A063496
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%I A063496
%S A063496 1,19,85,231,489,891,1469,2255,3281,4579,6181,8119,10425,13131,
%T A063496 16269,19871,23969,28595,33781,39559,45961,53019,60765,69231,78449,
%U A063496 88451,99269,110935,123481,136939,151341,166719,183105,200531,219029
%N A063496 (2*n-1)*(8*n^2-8*n+3)/3.
%C A063496 Number of potential flows in a 2 X 2 matrix with integer velocities in 
               -n..n, i.e. number of 2 X 2 matrices with adjacent elements differing 
               by no more than n, counting matrices differing by a constant only 
               once. - Ron Hardin (rhhardin(AT)att.net), Feb 27 2002
%C A063496 Number of ordered quadruples (a,b,c,d), -(n-1)<= a,b,c,d<=n-1, such that 
               a+b+c+d=0 - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 14 2003
%C A063496 If Y and Z are 2-blocks of a (2n+1)-set X then a(n-1) is the number of 
               5-subsets of X intersecting both Y and Z. - Milan R. Janjic (agnus(AT)blic.net), 
               Oct 28 2007
%C A063496 Equals binomial transform of [1, 18, 48, 32, 0, 0, 0,...]. - Gary W. 
               Adamson (qntmpkt(AT)yahoo.com), Jul 19 2008
%D A063496 T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. 
               (10).
%H A063496 Harry J. Smith, <a href="b063496.txt">Table of n, a(n) for n=1,...,1000</
               a>
%H A063496 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A063496 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%H A063496 R. Bacher, P. de la Harpe and B. Venkov, <a href="http://archive.numdam.org/
               ARCHIVE/AIF/AIF_1999__49_3/AIF_1999__49_3_727_0/AIF_1999__49_3_727_0.pdf">
               Series de croissance et series d'Ehrhart associees aux reseaux de 
               racines</a>, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
%F A063496 Comments from Peter Bala (pbala(AT)toucansurf.com), Jul 18 2008 (Start): 
               Partial sums of A010006. So this sequence is the crystal ball sequence 
               for the C_3 lattice - row 3 of A142992. The lattice C_3 consists 
               of all integer lattice points v = (a,b,c) in Z^3 such that a + b 
               + c is even, equipped with the taxicab type norm ||v|| = 1/2 * (|a| 
               + |b| + |c|).
%F A063496 The crystal ball sequence of C_3 gives the number of lattice points v 
               in C_3 with ||v|| <= n for n = 0,1,2,3,... [Bacher et al.].
%F A063496 For example, a(1) = 19 because the origin has norm 0 and the 18 lattice 
               points in Z^3 of norm 1 (as defined above) are +-(2,0,0), +-(0,2,
               0), +-(0,0,2), +-(1,1,0), +-(1,0,1), +-(0,1,1), +-(1,-1,0), +-(1,
               0,-1) and +-(0,1,-1). These 18 vectors form a root system of type 
               C_3.
%F A063496 O.g.f.: x*(1+15*x+15*x^2+x^3)/(1-x)^4 = x/(1-x) * T(3,(1+x)/(1-x)), where 
               T(n,x) denotes the Chebyshev polynomial of the first kind.
%F A063496 2*log(2) = 4/3 + sum {n = 1..inf} 1/(n*a(n)*a(n+1)). (End)
%o A063496 (PARI) { for (n=1, 1000, write("b063496.txt", n, " ", (2*n - 1)*(8*n^2 
               - 8*n + 3)/3) ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), 
               Aug 23 2009]
%Y A063496 1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, 
               A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, 
               A063492, A005917, A063493, A063494, A063495, A063496.
%Y A063496 Cf. A003215.
%Y A063496 Cf. A010006, A142992, A142993, A142994 .
%Y A063496 Sequence in context: A036564 A062639 A039609 this_sequence A027848 A039454 
               A142089
%Y A063496 Adjacent sequences: A063493 A063494 A063495 this_sequence A063497 A063498 
               A063499
%K A063496 nonn
%O A063496 1,2
%A A063496 N. J. A. Sloane (njas(AT)research.att.com), Aug 01 2001

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