A192780 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+1. See Comments.
1, 0, 1, 1, 2, 5, 8, 19, 34, 71, 137, 272, 537, 1056, 2089, 4112, 8121, 16009, 31586, 62301, 122888, 242411, 478146, 943183, 1860433, 3669792, 7238769, 14278720, 28165265, 55556896, 109587889, 216165713, 426394178, 841076725, 1659052040
Offset: 1
Keywords
Examples
The first five polynomials p(n,x) and their reductions: F1(x)=1 -> 1 F2(x)=x -> x F3(x)=x^2+1 -> x^2+1 F4(x)=x^3+2x -> x^2+2x+1 F5(x)=x^4+3x^2+1 -> 4x^2+1x+2, so that A192777=(1,0,1,1,2,...), A192778=(0,1,0,2,1,...), A192779=(0,0,1,1,4,...)
Links
- Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-3,1,1).
Programs
-
Mathematica
q = x^3; s = x^2 + 1; z = 40; p[n_, x_] := Fibonacci[n, x]; Table[Expand[p[n, x]], {n, 1, 7}] reduce[{p1_, q_, s_, x_}] := FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1] t = Table[reduce[{p[n, x], q, s, x}], {n, 1, z}]; u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192780 *) u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192781 *) u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}] (* A192782 *) LinearRecurrence[{1,3,-1,-3,1,1},{1,0,1,1,2,5},40] (* Harvey P. Dale, Nov 07 2021 *)
Formula
a(n)=a(n-1)+3*a(n-2)-a(n-3)-3*a(n-4)+a(n-5)+a(n-6).
G.f.: -x*(x-1)*(1+x)*(x^2+x-1) / ( -1+x+3*x^2-x^3-3*x^4+x^5+x^6 ). - R. J. Mathar, May 06 2014
Comments