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%I #72 Mar 15 2024 07:26:03
%S 1,19,85,231,489,891,1469,2255,3281,4579,6181,8119,10425,13131,16269,
%T 19871,23969,28595,33781,39559,45961,53019,60765,69231,78449,88451,
%U 99269,110935,123481,136939,151341,166719,183105,200531,219029
%N a(n) = (2*n - 1)*(8*n^2 - 8*n + 3)/3.
%C 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. - _R. H. Hardin_, Feb 27 2002
%C 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_, Jun 14 2003
%C 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 Janjic_, Oct 28 2007
%C Equals binomial transform of [1, 18, 48, 32, 0, 0, 0, ...]. - _Gary W. Adamson_, Jul 19 2008
%H Harry J. Smith, <a href="/A063496/b063496.txt">Table of n, a(n) for n = 1..1000</a>
%H 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">Séries de croissance et séries d'Ehrhart associées aux réseaux de racines</a>, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H T. P. Martin, <a href="http://dx.doi.org/10.1016/0370-1573(95)00083-6">Shells of atoms</a>, Phys. Rep., 273 (1996), 199-241, eq. (10).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F From _Peter Bala_, Jul 18 2008: (Start)
%F The following remarks about the C_3 lattice assume the sequence offset is 0.
%F 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 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 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 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 2*log(2) = 4/3 + Sum_{n >= 1} 1/(n*a(n)*a(n+1)). (End)
%F a(n+1) = (1/Pi) * Integral_{x=0..Pi} (sin((n+1/2)*x)/sin(x/2))^4. - _Yalcin Aktar_, Nov 02 2011, corrected by _R. J. Mathar_, Dec 01 2011
%F From _G. C. Greubel_, Dec 01 2017: (Start)
%F G.f.: x*(1 + 15*x + 15*x^2 + x^3)/(1 - x)^4.
%F E.g.f.: (-3 + 6*x + 24*x^2 + 16*x^3)*exp(x)/3 + 1. (End)
%F a(n) = A005900(2n-1). - _Ivan N. Ianakiev_, Mar 27 2022
%F From _Peter Bala_, Mar 11 2024: (Start)
%F Sum_{k = 1..n+1} 1/(k*a(k)*a(k+1)) = 1/(19 - 3/(27 - 60/(43 - 315/(67 - ... -n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*3^2))))).
%F E.g.f.: exp(x)*(1 + 18*x + 48*x^2/2! + 32*x^3/3!). Note that -T(6, i*sqrt(x)) = 1 + 18*x + 48*x^2 + 32*x^3, where T(n, x) denotes the n-th Chebyshev polynomial of the first kind. See A008310. (End)
%p A063496:=n->(2*n-1)*(8*n^2-8*n+3)/3; seq(A063496(n), n=1..40); # _Wesley Ivan Hurt_, May 09 2014
%t Table[(2*n - 1)*(8*n^2 - 8*n + 3)/3, {n, 40}] (* _Wesley Ivan Hurt_, May 09 2014 *)
%t LinearRecurrence[{4,-6,4,-1}, {1,19,85,231}, 30] (* _G. C. Greubel_, Dec 01 2017 *)
%o (PARI) { for (n=1, 1000, write("b063496.txt", n, " ", (2*n - 1)*(8*n^2 - 8*n + 3)/3) ) } \\ _Harry J. Smith_, Aug 23 2009
%o (PARI) x='x+O('x^30); Vec(serlaplace((-3+6*x+24*x^2+16*x^3)*exp(x)/3 + 1)) \\ _G. C. Greubel_, Dec 01 2017
%o (Magma) [(2*n-1)*(8*n^2-8*n+3)/3: n in [1..40]]; // _Wesley Ivan Hurt_, May 09 2014
%Y (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 Cf. A003215, A010006, A142992, A142993, A142994.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_, Aug 01 2001
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