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 A192772 Constant term in the reduction of the n-th Fibonacci polynomial by x^3->x^2+2x+1. 6
 1, 0, 1, 1, 2, 7, 12, 41, 86, 247, 585, 1548, 3849, 9896, 25001, 63724, 161721, 411257, 1044878, 2655719, 6748972, 17151849, 43589578, 110777391, 281529169, 715471992, 1818293377, 4620978640, 11743694657, 29845241080, 75848270001 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS For discussions of polynomial reduction, see A192232 and A192744. LINKS Index entries for linear recurrences with constant coefficients, signature (1,5,-1,-5,1,1). FORMULA a(n)=a(n-1)+5*a(n-2)-a(n-3)-5*a(n-4)+a(n-5)+a(n-6). G.f.: -x*(x^2-x-1)*(x^2+2*x-1) / (x^6+x^5-5*x^4-x^3+5*x^2+x-1). [Colin Barker, Jan 17 2013] EXAMPLE The first five polynomials p(n,x) and their reductions are as follows: F1(x)=1 -> 1 F2(x)=x -> x F3(x)=x^2+1 -> x^2+1 F4(x)=x^3+2x -> x^2+4x+1 F5(x)=x^4+3x^2+1 -> 6x^2+3x+2, so that A192772=(1,0,1,1,2,...), A192773=(0,1,0,4,3,...), A192774=(0,0,1,1,6,...) MATHEMATICA q = x^3; s = x^2 + 2 x + 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}]   (* A192772 *) u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]   (* A192773 *) u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}]   (* A192774 *) CROSSREFS Cf. A192744, A192232, A192616, A192773, A192774. Sequence in context: A055257 A238366 A084068 * A046243 A230302 A230637 Adjacent sequences:  A192769 A192770 A192771 * A192773 A192774 A192775 KEYWORD nonn,easy AUTHOR Clark Kimberling, Jul 09 2011 STATUS approved

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Last modified October 16 03:15 EDT 2019. Contains 328038 sequences. (Running on oeis4.)