A133585 Expansion of x - x^2*(2*x+1)*(x^2-2) / ( (x^2-x-1)*(x^2+x-1) ).

1, 2, 4, 5, 10, 13, 26, 34, 68, 89, 178, 233, 466, 610, 1220, 1597, 3194, 4181, 8362, 10946, 21892, 28657, 57314, 75025, 150050, 196418, 392836, 514229, 1028458, 1346269, 2692538, 3524578, 7049156, 9227465, 18454930, 24157817

Offset: 1

Author

Gary W. Adamson, Sep 18 2007

Keywords

• nonn
• easy

Comments

A133585 is a companion to A133586.

Examples

			a(4) = F(5) = 5.
a(5) = 2*a(4) = 2*5 = 10.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Crossrefs

Cf. A133080, A000045, A133586.

Programs

Maple

A133585aux := proc(n,k)
    add(A133566(n,j)*A133080(j,k),j=k..n) ;
end proc:
A000045 := proc(n)
    combinat[fibonacci](n) ;
end proc:
A133585 := proc(n)
    add(A133585aux(n,j)*A000045(j),j=0..n) ;
end proc: # R. J. Mathar, Jun 20 2015

Mathematica

CoefficientList[Series[1 - x (2 x + 1) (x^2 - 2)/((x^2 - x - 1) (x^2 + x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 21 2015 *)
LinearRecurrence[{0,3,0,-1},{1,2,4,5,10},40] (* Harvey P. Dale, Mar 04 2019 *)

PARI

a(n)=if(n>1,([0,1,0,0;0,0,1,0;0,0,0,1;-1,0,3,0]^(n-2)*[2;4;5;10])[1,1],1) \\ Charles R Greathouse IV, Jun 20 2015

Formula

Equals the matrix-matrix-vector product A133566 * A133080 * A000045 (previous name).

For even-indexed terms, a(n) = F(n+1). For odd-indexed terms (n>1), a(n) = 2*a(n-1), A126358.

Extensions

Previous name corrected and new name from R. J. Mathar, Jun 20 2015