A192430 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
1, 1, 3, 9, 33, 113, 403, 1409, 4977, 17489, 61619, 216809, 763377, 2686881, 9458787, 33295297, 117206177, 412579681, 1452347043, 5112464521, 17996645761, 63350804881, 223004208243, 785007489729, 2763341973393, 9727369663793
Offset: 0
Keywords
Examples
The first five polynomials p(n,x) and their reductions are as follows: p(0,x)=1 -> 1 p(1,x)=1+x -> 1+x p(2,x)=2+3x+x^2 -> 3+4x p(3,x)=2+7x+6x^2+x^3 -> 9+15x p(4,x)=4+12x+17x^2+10x^3+x^4 -> 33+52x. From these, read A192430=(1,1,3,9,33,...) and A192431=(0,1,4,15,52,...).
Programs
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Mathematica
q[x_] := x + 1; d = Sqrt[x + 2]; u[x_] := x + d; v[x_] := x - d; p[n_, x_] := (u[x]^n + v[x]^n)/2 + (u[x]^n - v[x]^n)/(2 d) (* A163762 *) Table[Expand[p[n, x]], {n, 0, 6}] reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)}; t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 0, 30}] Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192430 *) Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192431 *)
Formula
Conjecture: a(n) = 2*a(n-1)+6*a(n-2)-2*a(n-3)-a(n-4). G.f.: -(x^3+5*x^2+x-1) / (x^4+2*x^3-6*x^2-2*x+1). - Colin Barker, May 12 2014
Comments