A271388 a(n) = 4*a(n-1) + a(n-2) - n for n > 1, with a(0) = 0, a(1) = 1.

0, 1, 2, 6, 22, 89, 372, 1570, 6644, 28137, 119182, 504854, 2138586, 9059185, 38375312, 162560418, 688616968, 2917028273, 12356730042, 52343948422, 221732523710, 939274043241, 3978828696652, 16854588829826, 71397184015932, 302443324893529, 1281170483590022

Offset: 0

Author

Ilya Gutkovskiy, Apr 06 2016

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (6,-8,2,1).

Crossrefs

Cf. A030119, A033887, A048776, A098317.

Programs

Mathematica

RecurrenceTable[{a[0] == 0, a[1] == 1, a[n] == 4 a[n - 1] + a[n - 2] - n}, a, {n, 28}]
LinearRecurrence[{6, -8, 2, 1}, {0, 1, 2, 6}, 29]
nxt[{n_,a_,b_}]:={n+1,b,4b+a-n-1}; NestList[nxt,{1,0,1},30][[;;,2]] (* Harvey P. Dale, Feb 07 2025 *)

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

PARI

a(n) = (3*fibonacci(3*n-2) + 2*n+3) >> 3; \\ Kevin Ryde, May 16 2021

Formula

G.f.: x*(1 - 4*x + 2*x^2)/((1 - x)^2*(1 - 4*x - x^2)).

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)).

a(n) = 6*a(n-1) - 8*a(n-2) + 2*a(n-3) + a(n-4).

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).

Lim_{n->infinity} a(n + 1)/a(n) = 2 + sqrt(5) = phi^3 = A098317, where phi is the golden ratio (A001622).

a(n) = (2*n + 3 + 3*A033887(n-1))/8. - R. J. Mathar, Mar 12 2017