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

Author

Michael Somos, Mar 11 2004

Keywords

• nonn
• easy

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).

Examples

			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.
		

Links

• Index entries for two-way infinite sequences
• Index entries for linear recurrences with constant coefficients, signature (1,2,1,-1).

Programs

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 *)

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])))

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).