%I #26 May 07 2024 08:05:41
%S 1,1,5,9,13,25,25,49,41,81,61,121,85,169,113,225,145,289,181,361,221,
%T 441,265,529,313,625,365,729,421,841,481,961,545,1089,613,1225,685,
%U 1369,761,1521,841,1681,925,1849,1013,2025,1105,2209,1201,2401,1301,2601,1405
%N Expansion of (1 + x + 2*x^2 + 6*x^3 + x^4 + x^5)/(1 - x^2)^3.
%C Centered square numbers alternating with odd squares.
%H Jinyuan Wang, <a href="/A271391/b271391.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).
%F E.g.f.: ((2 + x*(2 + x))*cosh(x) + x*(3 + 2*x)*sinh(x))/2.
%F a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).
%F a(n) = (3*n^2 + 2*n + 2 + (-1)^n*(-n^2 + 2*n + 2))/4.
%F a(2n) = A001844(n). a(2n+1) = (2n+1)^2.
%e Illustration of initial terms:
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%e 0 1 2 3 4 5 6
%p a:=series((1+x+2*x^2+6*x^3+x^4+x^5)/(1-x^2)^3,x=0,55): seq(coeff(a,x,n),n=0..54); # _Paolo P. Lava_, Mar 27 2019
%t LinearRecurrence[{0, 3, 0, -3, 0, 1}, {1, 1, 5, 9, 13, 25}, 55]
%t Table[(3 n^2 + 2 n + 2 + (-1)^n (-n^2 + 2 n + 2))/4, {n, 0, 54}]
%o (PARI) x='x+O('x^99); Vec((1+x+2*x^2+6*x^3+x^4+x^5)/(1-x^2)^3) \\ _Altug Alkan_, Apr 06 2016
%Y Cf. A001844, A016754.
%K nonn,easy
%O 0,3
%A _Ilya Gutkovskiy_, Apr 06 2016