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

1,2

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).

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

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 *)

Prog

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

Crossrefs

Cf. A000045, A003215, A005891.

Sequence in context: A209995 A050410 A121991 * A050491 A332589 A022283

Adjacent sequences: A121987 A121988 A121989 * A121991 A121992 A121993

Keyword

nonn

Author

Roger L. Bagula, Sep 10 2006

Extensions

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

Status

approved