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%I #61 Sep 08 2022 12:38:20
%S 1,1,3,5,9,13,20,28,39,51,67,85,107,131,160,192,229,269,315,365,421,
%T 481,548,620,699,783,875,973,1079,1191,1312,1440,1577,1721,1875,2037,
%U 2209,2389,2580,2780,2991,3211,3443,3685,3939,4203,4480,4768,5069,5381,5707,6045
%N Expansion of (1+t^3)^2/((1-t)*(1-t^2)^2*(1-t^4)).
%C Molien series for 5-dimensional group of order 8.
%C For of each of the quadrisections the n-th term is a polynomial in n of degree 3. - _Ralf Stephan_, Nov 16 2010
%C Number of non-isomorphic 3 X 3 nonnegative integer matrices with all row and column sums equal to n up to permutations of rows and columns. - _Andrew Howroyd_, Apr 08 2020
%C Take the square spiral on the square grid, with cells on the spiral numbered starting at 1. Every time the spiral crosses the x- or y-axis, calculate the sum of the numbers on the intersection of the spiral and the axis. This produces the present sequence (see illustration). - _Karl-Heinz Hofmann_, Aug 14 2022
%H G. C. Greubel, <a href="/A106607/b106607.txt">Table of n, a(n) for n = 0..1000</a>
%H S. Ling and P. Solé, <a href="https://doi.org/10.1006/eujc.2001.0509">Type II Codes over F_4 + u F_4</a>, European J. Combinatorics, 22 (2001), pp. 983-997.
%H Karl-Heinz Hofmann, <a href="/A106607/a106607_1.pdf">An alternative way to get the terms of A106607. Examples of a(1..18)</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,1,-3,3,-1).
%F G.f.: (1-x+x^2)^2/( (1+x)*(1+x^2)*(1-x)^4 ). - _R. J. Mathar_, Dec 18 2014
%F a(n) = (4*n^3 +18*n^2 +56*n +3*(9*(-1)^n +2*(1-i)*(-i)^n +2*(1+i)*i^n +19))/96 where i is the imaginary unit. - _Colin Barker_, Feb 08 2016
%F E.g.f.: (1/48)*(6*(cos(x) - sin(x)) + p(x)*sinh(x) + (27 + p(x))*cosh(x)), where p(x) = 15 + 39*x + 15*x^2 + 2*x^3. - _G. C. Greubel_, Sep 08 2021
%e The a(4) = 9 symmetric matrices are:
%e [0 0 4] [0 1 3] [0 1 3] [0 2 2] [0 2 2]
%e [0 4 0] [1 2 1] [1 3 0] [2 0 2] [2 1 1]
%e [4 0 0] [3 1 0] [3 0 1] [2 2 0] [2 1 1]
%e .
%e [1 1 2] [1 0 3] [1 1 2] [2 0 2]
%e [1 2 1] [0 4 0] [1 3 0] [0 4 0]
%e [2 1 1] [3 0 1] [2 0 2] [2 0 2]
%p (1+t^3)^2/((1-t)*(1-t^2)^2*(1-t^4));
%p seq(coeff(series(%,t,n+1), t,n), n=0..60);
%t LinearRecurrence[{3,-3,1,1,-3,3,-1}, {1,1,3,5,9,13,20}, 61] (* _G. C. Greubel_, Sep 08 2021 *)
%o (PARI) a(n) = i=I; (4*n^3+18*n^2+56*n+3*(9*(-1)^n+(2-2*i)*(-i)^n+(2+2*i)*i^n+19))/96 \\ _Colin Barker_, Feb 08 2016
%o (Sage)
%o def A106607_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1+x^3)^2/((1-x)*(1-x^2)^2*(1-x^4)) ).list()
%o A106607_list(60) # _G. C. Greubel_, Sep 08 2021
%Y Row n=3 of A333737.
%Y Cf. A100779.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, May 12 2005
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