A192350 - OEIS

%I #7 Jan 17 2013 09:05:09

%S 1,0,6,4,64,128,896,2752,14208,52224,238592,946176,4110336,16830464,

%T 71598080,297140224,1253048320,5229707264,21973303296,91924463616,

%U 385642135552,1614916091904,6770569248768,28364203098112,118885634277376

%N Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

%C To define the polynomials p(n,x), let d=sqrt(x+5); then p(n,x)=(1/2)((x+d)^n+(x-d)^n).  These are similar to polynomials at A161516.

%C For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

%F Conjecture: a(n) = 2*a(n-1)+12*a(n-2)-8*a(n-3)-16*a(n-4). G.f.: -x*(6*x^2+2*x-1) / (16*x^4+8*x^3-12*x^2-2*x+1). [_Colin Barker_, Jan 17 2013]

%e The first four polynomials p(n,x) and their reductions are as follows:

%e p(0,x)=1 -> 1

%e p(1,x)=x -> x

%e p(2,x)=5+x+x^2 -> 6+2x

%e p(3,x)=15x+3x^2+x^3 -> 4+20x.

%e From these, we read

%e A192350=(1,0,6,4,...) and A192351=(0,1,2,20...)

%t q[x_] := x + 1; d = Sqrt[x + 5];

%t p[n_, x_] := ((x + d)^n + (x - d)^n )/

%t   2 (* similar to polynomials defined at A161516 *)

%t Table[Expand[p[n, x]], {n, 0, 4}]

%t reductionRules = {x^y_?EvenQ -> q[x]^(y/2),

%t    x^y_?OddQ -> x q[x]^((y - 1)/2)};

%t t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]

%t Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]

%t   (* A192350 *)

%t Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]

%t   (* A192351 *)

%Y Cf. A192232, A192351.

%K nonn

%O 1,3

%A _Clark Kimberling_, Jun 28 2011