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A192468 Constant term of the reduction by x^2->x+3 of the polynomial p(n,x)=1+x^n+x^(2n).

%I A192468 #7 Nov 12 2012 09:55:04
%S A192468 4,16,61,304,1546,8107,42748,226240,1198645,6353944,33688474,
%T A192468 178631251,947215924,5022815920,26634734125,141237718720,748951245034,
%U A192468 3971518837243,21060069709228,111676816254688,592197081386533,3140288211876136
%N A192468 Constant term of the reduction by x^2->x+3 of the polynomial p(n,x)=1+x^n+x^(2n).
%C A192468 For an introduction to reductions of polynomials by substitutions such as x^2->x+3, see A192232.
%F A192468 Empirical G.f.: -x*(81*x^4-87*x^3-x^2+20*x-4)/((x-1)*(3*x^2+x-1)*(9*x^2-7*x+1)). [_Colin Barker_, Nov 12 2012]
%e A192468 The first four polynomials p(n,x) and their reductions are as follows:
%e A192468 p(1,x)=1+x+x^2 -> 4+2x
%e A192468 p(2,x)=1+x^2+x^4 -> 16+8x
%e A192468 p(3,x)=1+x^3+x^6 -> 61+44x
%e A192468 p(4,x)=1+x^4+x^8 -> 304+224x.
%e A192468 From these, read
%e A192468 A192468=(4,16,61,304,...) and A192469=(2,8,44,224,...)
%t A192468 Remove["Global`*"];
%t A192468 q[x_] := x + 3; p[n_, x_] := 1 + x^n + x^(2 n);
%t A192468 Table[Simplify[p[n, x]], {n, 1, 5}]
%t A192468 reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t A192468    x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192468 t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}]
%t A192468 Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
%t A192468 (* A192468 *)
%t A192468 Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
%t A192468 (* A192469 *)
%t A192468 Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]
%t A192468 (* A192470 *)
%Y A192468 Cf. A192232, A192468, A192464, A192465.
%K A192468 nonn
%O A192468 1,1
%A A192468 _Clark Kimberling_, Jul 01 2011