%I #14 May 16 2021 01:48:59
%S 0,1,6,22,66,179,460,1148,2820,6869,16658,40306,97414,235303,568216,
%T 1371960,3312392,7997033,19306782,46610958,112529098,271669595,
%U 655868772,1583407668,3822684684,9228777661,22280240682,53789259754,129858760974,313506782543,756872326960
%N a(n) = 2*a(n-1) + a(n-2) + n^2 for n > 1, with a(0) = 0, a(1) = 1.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-8,4,1,-1).
%F G.f.: x*(1 + x)/((1 - x)^3*(1 - 2*x - x^2)).
%F E.g.f.: (1/4)*exp(x)*(-2*(x*(x + 5) + 5) + 7*sqrt(2)*sinh(sqrt(2)*x) + 10*cosh(sqrt(2)*x)).
%F a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3) + a(n-4) - a(n-5).
%F a(n) = (1/8)*(-4*(n*(n + 4) + 5) + (10 - 7*sqrt(2))*(1 - sqrt(2))^n + (10 + 7*sqrt(2))*(1 + sqrt(2))^n).
%F Lim_{n->infinity} a(n + 1)/a(n) = 1 + sqrt(2) = A014176.
%F a(n) = (A000129(n+3) - A002522(n+2))/2. - _R. J. Mathar_, Jun 07 2016
%t RecurrenceTable[{a[0] == 0, a[1] == 1, a[n] == 2 a[n - 1] + a[n - 2] + n^2}, a, {n, 31}]
%t LinearRecurrence[{5, -8, 4, 1, -1}, {0, 1, 6, 22, 66}, 32]
%o (PARI) x='x+O('x^99); concat(0, Vec(x*(1+x)/((1-x)^3*(1-2*x-x^2)))) \\ _Altug Alkan_, Apr 06 2016
%Y Cf. A000129, A002522, A014176, A053808.
%K nonn,easy
%O 0,3
%A _Ilya Gutkovskiy_, Apr 06 2016