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A192423 Constant term of the reduction by x^2->x+2 of the polynomial p(n,x) defined below in Comments. 3

%I

%S 2,0,4,2,16,20,78,140,416,878,2324,5280,13282,31200,76724,182962,

%T 445376,1069300,2591118,6239980,15089776,36389278,87917284,212144640,

%U 512334722,1236606720,2985883684,7207831202,17402424496,42011258900

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

%C The polynomial p(n,x) is defined by ((x+d)/2)^n+((x-d)/2)^n, where d=sqrt(x^2+4). For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.

%F Conjecture: a(n) = a(n-1)+4*a(n-2)-a(n-3)-a(n-4). G.f.: -2*(x+1)*(2*x-1) / ((x^2-x-1)*(x^2+2*x-1)). - _Colin Barker_, May 11 2014

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

%e p(0,x)=2 -> 2

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

%e p(2,x)=2+x^2 -> 4+x

%e p(3,x)=3x+x^3 -> 2+6x

%e p(4,x)=2+4x^2+x^4 -> 16+9x.

%e From these, read A192423=(2,0,4,2,16,...) and A192424=(0,1,1,6,9,...)

%t q[x_] := x + 2; d = Sqrt[x^2 + 4];

%t p[n_, x_] := ((x + d)/2)^n + ((x - d)/2)^n (* A161514 *)

%t Table[Expand[p[n, x]], {n, 0, 6}]

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

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

%t Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192423 *)

%t Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192424 *)

%t Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}] (* A192425 *)

%Y Cf. A192232, A192424.

%K nonn

%O 0,1

%A _Clark Kimberling_, Jun 30 2011

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Last modified September 25 07:13 EDT 2020. Contains 337335 sequences. (Running on oeis4.)