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%I #20 May 16 2021 03:04:34
%S 0,1,2,6,22,89,372,1570,6644,28137,119182,504854,2138586,9059185,
%T 38375312,162560418,688616968,2917028273,12356730042,52343948422,
%U 221732523710,939274043241,3978828696652,16854588829826,71397184015932,302443324893529,1281170483590022
%N a(n) = 4*a(n-1) + a(n-2) - n for n > 1, with a(0) = 0, a(1) = 1.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-8,2,1).
%F G.f.: x*(1 - 4*x + 2*x^2)/((1 - x)^2*(1 - 4*x - x^2)).
%F E.g.f.: (1/80)*(10*exp(x)*(2*x + 3) - 3*(5 + 3*sqrt(5))*exp((2 - sqrt(5))*x) + 3*(3*sqrt(5) - 5)*exp((2 + sqrt(5))*x)).
%F a(n) = 6*a(n-1) - 8*a(n-2) + 2*a(n-3) + a(n-4).
%F a(n) = (1/80)*(20*n - 3*(5 + 3*sqrt(5))*(2 - sqrt(5))^n + 3*(3*sqrt(5) - 5)*(2 + sqrt(5))^n + 30).
%F Lim_{n->infinity} a(n + 1)/a(n) = 2 + sqrt(5) = phi^3 = A098317, where phi is the golden ratio (A001622).
%F a(n) = (2*n + 3 + 3*A033887(n-1))/8. - _R. J. Mathar_, Mar 12 2017
%t RecurrenceTable[{a[0] == 0, a[1] == 1, a[n] == 4 a[n - 1] + a[n - 2] - n}, a, {n, 28}]
%t LinearRecurrence[{6, -8, 2, 1}, {0, 1, 2, 6}, 29]
%o (PARI) x='x+O('x^99); concat(0, Vec(x*(1-4*x+2*x^2)/((1-x)^2*(1-4*x-x^2)))) \\ _Altug Alkan_, Apr 06 2016
%o (PARI) a(n) = (3*fibonacci(3*n-2) + 2*n+3) >> 3; \\ _Kevin Ryde_, May 16 2021
%Y Cf. A030119, A033887, A048776, A098317.
%K nonn,easy
%O 0,3
%A _Ilya Gutkovskiy_, Apr 06 2016
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