A105371 Expansion of (1-x)*(1-x+x^2)/(1-3*x+4*x^2-2*x^3+x^4).

1, 1, 1, 0, -3, -8, -13, -13, 0, 34, 89, 144, 144, 0, -377, -987, -1597, -1597, 0, 4181, 10946, 17711, 17711, 0, -46368, -121393, -196418, -196418, 0, 514229, 1346269, 2178309, 2178309, 0, -5702887, -14930352, -24157817, -24157817, 0, 63245986, 165580141

Offset: 0

Author

Paul Barry, Apr 01 2005

Keywords

• easy
• sign

Comments

Binomial transform of A105367.

Links

• Index entries for two-way infinite sequences
• Index entries for linear recurrences with constant coefficients, signature (3,-4,2,-1).

Crossrefs

Cf. A000045, A105367.

Programs

Mathematica

CoefficientList[Series[(1-x)(1-x+x^2)/(1-3x+4x^2-2x^3+x^4),{x,0,60}],x] (* or *) LinearRecurrence[{3,-4,2,-1},{1,1,1,0},60] (* Harvey P. Dale, Dec 21 2013 *)

PARI

a(n)=local(m); m=n%5+1; [1,-1,-1,0,1][m]*fibonacci(-n-(m<3)) /* Michael Somos, Apr 09 2005 */

Formula

G.f.: (1-2x+2x^2-x^3)/(1-3x+4x^2-2x^3+x^4).

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

a(n) = (1/2+sqrt(5)/2)^n((1/2+sqrt(5)/10)cos(Pi*n/5)+sqrt(1/10-sqrt(5)/50)sin(Pi*n/5))- (sqrt(5)/2-1/2)^n((sqrt(5)/10-1/2)cos(2*Pi*n/5)+sqrt(1/10+sqrt(5)/50)sin(2*Pi*n/5)).

a(5n) = -F(-5n-1), a(5n+1) = a(5n+2) = -F(-5n-2), a(5n+3) = 0, a(5n+4) = F(-5n-4). - Michael Somos, Apr 09 2005