cp's OEIS Frontend

A192465 Constant term of the reduction by x^2->x+2 of the polynomial p(n,x)=1+x^n+x^(2n).

%I A192465 #8 Nov 12 2012 09:13:24
%S A192465 3,9,25,93,353,1389,5505,21933,87553,349869,1398785,5593773,22372353,
%T A192465 89483949,357924865,1431677613,5726666753,22906579629,91626143745,
%U A192465 366504225453,1466016202753,5864063412909,23456250855425,93824997829293
%N A192465 Constant term of the reduction by x^2->x+2 of the polynomial p(n,x)=1+x^n+x^(2n).
%C A192465 For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
%F A192465 Empirical G.f.: -x*(3*x-1)*(8*x^2-3)/((x-1)*(x+1)*(2*x-1)*(4*x-1)). [_Colin Barker_, Nov 12 2012]
%e A192465 The first four polynomials p(n,x) and their reductions are as follows:
%e A192465 p(1,x)=1+x+x^2 -> 3+2x
%e A192465 p(2,x)=1+x^2+x^4 -> 9+6x
%e A192465 p(3,x)=1+x^3+x^6 -> 25+24x
%e A192465 p(4,x)=1+x^4+x^8 -> 93+90x.
%e A192465 From these, read
%e A192465 A192465=(3,9,25,93,...) and A192466=(2,6,24,90,...)
%t A192465 Remove["Global`*"];
%t A192465 q[x_] := x + 2; p[n_, x_] := 1 + x^n + x^(2 n);
%t A192465 Table[Simplify[p[n, x]], {n, 1, 5}]
%t A192465 reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t A192465    x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192465 t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}]
%t A192465 Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
%t A192465 (* A192465 *)
%t A192465 Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
%t A192465 (* A192466 *)
%t A192465 Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]
%t A192465 (* A192467 *)
%Y A192465 Cf. A192232, A192466, A192464.
%K A192465 nonn
%O A192465 1,1
%A A192465 _Clark Kimberling_, Jul 01 2011