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A192472 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x)=1+x^n+x^(2n+2).

%I A192472 #7 Nov 12 2012 09:54:41
%S A192472 3,7,15,37,93,239,619,1611,4203,10981,28713,75115,196563,514463,
%T A192472 1346647,3525189,9228453,24159415,63248571,165584323,433501203,
%U A192472 1134914117,2971232785,7778770707,20365057443,53316366199,139583983839,365435492581
%N A192472 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x)=1+x^n+x^(2n+2).
%C A192472 For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
%F A192472 Empirical G.f.: -x*(3*x^4-8*x^3-x^2+8*x-3)/((x-1)*(x^2-3*x+1)*(x^2+x-1)). [_Colin Barker_, Nov 12 2012]
%e A192472 The first four polynomials p(n,x) and their reductions are as follows:
%e A192472 p(1,x)=1+x+x^4 -> 3+4x
%e A192472 p(2,x)=1+x^2+x^6 -> 7+9x
%e A192472 p(3,x)=1+x^3+x^8 -> 15+23x
%e A192472 p(4,x)=1+x^4+x^10 -> 37+58x.
%e A192472 From these, read
%e A192472 A192472=(3,7,15,37,...) and A192473=(4,9,23,58,...)
%t A192472 Remove["Global`*"];
%t A192472 q[x_] := x + 1; p[n_, x_] := 1 + x^n + x^(2 n+2);
%t A192472 Table[Simplify[p[n, x]], {n, 1, 5}]
%t A192472 reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t A192472    x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192472 t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}]
%t A192472 Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
%t A192472 (* A192472 *)
%t A192472 Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
%t A192472 (* A192473 *)
%Y A192472 Cf. A192232, A192473.
%K A192472 nonn
%O A192472 1,1
%A A192472 _Clark Kimberling_, Jul 01 2011