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A192465 Constant term of the reduction by x^2->x+2 of the polynomial p(n,x)=1+x^n+x^(2n). 4

%I

%S 3,9,25,93,353,1389,5505,21933,87553,349869,1398785,5593773,22372353,

%T 89483949,357924865,1431677613,5726666753,22906579629,91626143745,

%U 366504225453,1466016202753,5864063412909,23456250855425,93824997829293

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

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

%F 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 The first four polynomials p(n,x) and their reductions are as follows:

%e p(1,x)=1+x+x^2 -> 3+2x

%e p(2,x)=1+x^2+x^4 -> 9+6x

%e p(3,x)=1+x^3+x^6 -> 25+24x

%e p(4,x)=1+x^4+x^8 -> 93+90x.

%e From these, read

%e A192465=(3,9,25,93,...) and A192466=(2,6,24,90,...)

%t Remove["Global`*"];

%t q[x_] := x + 2; p[n_, x_] := 1 + x^n + x^(2 n);

%t Table[Simplify[p[n, x]], {n, 1, 5}]

%t reductionRules = {x^y_?EvenQ -> q[x]^(y/2),

%t x^y_?OddQ -> x q[x]^((y - 1)/2)};

%t t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}]

%t Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]

%t (* A192465 *)

%t Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]

%t (* A192466 *)

%t Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]

%t (* A192467 *)

%Y Cf. A192232, A192466, A192464.

%K nonn

%O 1,1

%A _Clark Kimberling_, Jul 01 2011

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Last modified September 27 17:05 EDT 2021. Contains 347693 sequences. (Running on oeis4.)