A092886 Expansion of x/(x^4-x^3-2x^2-x+1).

0, 1, 1, 3, 6, 12, 26, 53, 111, 231, 480, 1000, 2080, 4329, 9009, 18747, 39014, 81188, 168954, 351597, 731679, 1522639, 3168640, 6594000, 13722240, 28556241, 59426081, 123666803, 257352966, 535556412, 1114503066, 2319302053

Offset

0,4

Comments

If P(x),Q(x) are n-th and (n-1)-th Fibonacci polynomials, then a(n)=real part of the product of P(I) and conjugate Q(I).

Links

Table of n, a(n) for n=0..31.

Index entries for two-way infinite sequences

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

Formula

G.f.: x/(x^4-x^3-2x^2-x+1). a(n)=a(n-1)+2*a(n-2)+a(n-3)-a(n-4). a(n)=-a(-2-n).

Example

Fibonacci polynomials P(5)=1+4x+3x^2, P(4)=1+3x+x^2. Conjugate product evaluated at I is (-2+4I)*(-3I)=12-6I and so a(5)=12.

Mathematica

CoefficientList[Series[x/(x^4-x^3-2x^2-x+1), {x, 0, 40}], x] (* or *) LinearRecurrence[{1, 2, 1, -1}, {0, 1, 1, 3}, 40] (* Harvey P. Dale, Feb 27 2015 *)

Prog

(PARI) a(n)=local(m); if(n<1, if(n>-3, 0, -a(-2-n)), m=contfracpnqn(matrix(2, n, i, j, I)); real(m[1, 1]*conj(m[2, 1])))

Crossrefs

Sequence in context: A278821 A274476 A274059 * A207094 A135035 A054195

Adjacent sequences: A092883 A092884 A092885 * A092887 A092888 A092889

Keyword

nonn,easy

Author

Michael Somos, Mar 11 2004

Status

approved