A192346 Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.
1, 0, 3, 4, 25, 68, 275, 904, 3297, 11400, 40499, 141900, 500697, 1760396, 6200675, 21820432, 76823425, 270407696, 951914403, 3350807700, 11795463001, 41521535700, 146162319603, 514512119704, 1811159622625, 6375545788568, 22442862753875
Offset: 1
Keywords
Examples
The first four polynomials p(n,x) and their reductions are as follows: p(0,x)=1 -> 1 p(1,x)=x -> x p(2,x)=2+x+x^2 -> 3+2x p(3,x)=6x+3x^2+x^3 -> 4+11x. From these, we read A192346=(1,0,3,4,...) and A192347=(1,1,2,11...)
Programs
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Mathematica
q[x_] := x + 1; d = Sqrt[x + 2]; p[n_, x_] := ((x + d)^n + (x - d)^n )/ 2 (* similar to polynomials defined at A161516 *) Table[Expand[p[n, x]], {n, 0, 4}] reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)}; t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}] Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192346 *) Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192347 *)
Formula
Conjecture: a(n) = 2*a(n-1)+6*a(n-2)-2*a(n-3)-a(n-4). G.f.: -x*(x+1)*(3*x-1) / (x^4+2*x^3-6*x^2-2*x+1). [Colin Barker, Jan 17 2013]
Comments