A193046 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.
1, 1, 17, 83, 275, 727, 1673, 3505, 6873, 12843, 23155, 40639, 69889, 118353, 198097, 328659, 541667, 888311, 1451433, 2365089, 3846201, 6245771, 10131747, 16423103, 26606785, 43088737, 69761873, 112925075, 182770163, 295787863
Offset: 0
Links
- Index entries for linear recurrences with constant coefficients, signature (5,-9,6,1,-3,1).
Programs
-
Mathematica
q = x^2; s = x + 1; z = 40; p[0, x] := 1; p[n_, x_] := x*p[n - 1, x] + n^4; Table[Expand[p[n, x]], {n, 0, 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, 0, z}]; u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A193046 *) u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A193047 *)
Formula
a(n) = 5*a(n-1)-9*a(n-2)+6*a(n-3)+a(n-4)-3*a(n-5)+a(n-6).
G.f.: (x^5-6*x^4-x^3-21*x^2+4*x-1) / ((x-1)^4*(x^2+x-1)). - Colin Barker, May 11 2014
Comments