A192348 Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.
1, 0, 4, 4, 36, 88, 432, 1408, 5776, 20736, 80320, 297792, 1132096, 4242304, 16028928, 60276736, 227287296, 855703552, 3224482816, 12144337920, 45752574976, 172339107840, 649223532544, 2445572276224, 9212566081536, 34703459811328
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)=3+x+x^2 -> 4+2x p(3,x)=9x+3x^2+x^3 -> 4+14x. From these, we read A192348=(1,0,3,4,...) and A192349=(0,1,2,14...)
Programs
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Mathematica
q[x_] := x + 1; d = Sqrt[x + 3]; 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}] (* A192348 *) Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192349 *)
Formula
Conjecture: a(n) = 2*a(n-1)+8*a(n-2)-4*a(n-3)-4*a(n-4). G.f.: -x*(4*x^2+2*x-1) / (4*x^4+4*x^3-8*x^2-2*x+1). [Colin Barker, Jan 17 2013]
Comments