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A192353 Constant term of the reduction of the polynomial p(n,x)=(1/2)((x+2)^n+(x-2)^n) by x^2->x+1.

%I A192353 #10 Sep 12 2012 06:55:05
%S A192353 1,0,5,1,42,43,429,820,4861,12597,58598,177859,732825,2417416,9358677,
%T A192353 32256553,120902914,426440955,1571649221,5610955132,20497829133,
%U A192353 73645557469,267803779710,965384509651,3502058316337,12646311635088,45818284122149
%N A192353 Constant term of the reduction of the polynomial p(n,x)=(1/2)((x+2)^n+(x-2)^n) by x^2->x+1.
%C A192353 For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
%F A192353 Empirical G.f.: x*(x^3-4*x^2-2*x+1)/((x^2+3*x+1)*(5*x^2-5*x+1)). [_Colin Barker_, Sep 11 2012]
%e A192353 (See A192352 for a related example.)
%t A192353 q[x_] := x + 1; d = 2;
%t A192353 p[n_, x_] := ((x + d)^n + (x - d)^n )/2 (* similar to polynomials defined at A161516 *)
%t A192353 Table[Expand[p[n, x]], {n, 0, 6}]
%t A192353 reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t A192353    x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192353 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
%t A192353 Table[Coefficient[Part[t,n],x,0], {n,1,30}](* A192353 *)
%t A192353 Table[Coefficient[Part[t,n],x,1], {n,1,30}]  (* A192354 *)
%Y A192353 Cf. A192232, A192354, A192352.
%K A192353 nonn
%O A192353 1,3
%A A192353 _Clark Kimberling_, Jun 29 2011