A193004 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.
1, 1, 9, 29, 75, 165, 331, 623, 1123, 1963, 3357, 5651, 9405, 15525, 25477, 41633, 67831, 110281, 179031, 290339, 470511, 762111, 1234009, 1997639, 3233305, 5232745, 8468001, 13702853, 22173123, 35878413, 58054147, 93935351, 151992475
Offset: 0
Links
- Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).
Programs
-
Mathematica
q = x^2; s = x + 1; z = 40; p[0, x] := 1; p[n_, x_] := x*p[n - 1, x] + n^3; 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}] (* A193004 *) u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A193005 *) -
PARI
Vec((x^4-3*x^3+10*x^2-3*x+1)/((x-1)^3*(x^2+x-1)) + O(x^100)) \\ Colin Barker, May 12 2014
Formula
a(n) = 4*a(n-1)-5*a(n-2)+a(n-3)+2*a(n-4)-a(n-5).
G.f.: (x^4-3*x^3+10*x^2-3*x+1) / ((x-1)^3*(x^2+x-1)). - Colin Barker, May 12 2014
Comments