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

%I #10 Sep 12 2012 06:55:05

%S 1,0,5,1,42,43,429,820,4861,12597,58598,177859,732825,2417416,9358677,

%T 32256553,120902914,426440955,1571649221,5610955132,20497829133,

%U 73645557469,267803779710,965384509651,3502058316337,12646311635088,45818284122149

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

%F 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 (See A192352 for a related example.)

%t q[x_] := x + 1; d = 2;

%t p[n_, x_] := ((x + d)^n + (x - d)^n )/2 (* similar to polynomials defined at A161516 *)

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

%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}](* A192353 *)

%t Table[Coefficient[Part[t,n],x,1], {n,1,30}] (* A192354 *)

%Y Cf. A192232, A192354, A192352.

%K nonn

%O 1,3

%A _Clark Kimberling_, Jun 29 2011