A121990 Expansion of x*(1+9*x+2*x^2)/((1-x)*(1-3*x+x^2)).

1, 13, 50, 149, 409, 1090, 2873, 7541, 19762, 51757, 135521, 354818, 928945, 2432029, 6367154, 16669445, 43641193, 114254146, 299121257, 783109637, 2050207666, 5367513373, 14052332465, 36789484034, 96316119649, 252158874925

Offset: 1

Author

Roger L. Bagula, Sep 10 2006

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Crossrefs

Cf. A000045, A003215, A005891.

Programs

GAP

F:=Fibonacci;; List([1..30], n-> 12*F(2*n-1) +F(2*n-3) -12 ); # G. C. Greubel, Nov 21 2019

Magma

F:= Fibonacci; [12*F(2*n-1) +F(2*n-3) -12: n in [1..30]]; // G. C. Greubel, Nov 21 2019

Maple

with(combinat); seq(12*fibonacci(2*n-1) +fibonacci(2*n-3) -12, n=1..30); # G. C. Greubel, Nov 21 2019

Mathematica

LinearRecurrence[{4,-4,1}, {1,13,50}, 30] (* G. C. Greubel, Sep 14 2017 *)
With[{F=Fibonacci}, Table[12*(F[2*n-1]-1) + F[2*n-3], {n,30}]] (* G. C. Greubel, Nov 21 2019 *)

PARI

x='x+O('x^30); Vec(x*(1+9*x+2*x^2)/((1-x)*(x^2-3*x+1))) \\ G. C. Greubel, Sep 14 2017

PARI

vector(30, n, 12*fibonacci(2*n-1) +fibonacci(2*n-3) -12) \\ G. C. Greubel, Nov 21 2019

Sage

f=fibonacci; [12*f(2*n-1) + f(2*n-3) -12 for n in (1..30)] # G. C. Greubel, Nov 21 2019

Formula

a(n) = 3*a(n - 1) - a(n - 2) + 12.

a(n) = (1/10)*(-120 + (65 - 11*sqrt(5))*((1/2)*(3 - sqrt(5)))^n + ((1/2)*(3 + sqrt(5)))^n*(65 + 11*sqrt(5))).

From R. J. Mathar, Apr 04 2009: (Start)

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

G.f.: x*(1+9*x+2*x^2)/((1-x)*(1-3*x+x^2)). (End)

a(n) = 12*Fibonacci(2*n-1) + Fibonacci(2*n-3) - 12. - G. C. Greubel, Nov 21 2019

Extensions

Edited and new name based on g.f. by G. C. Greubel and Joerg Arndt, Sep 14 2017