cp's OEIS Frontend

A192466 Coefficient of x in the reduction by x^2->x+2 of the polynomial p(n,x)=1+x^n+x^(2n).

%I A192466 #16 Jul 07 2020 05:16:58
%S A192466 2,6,24,90,352,1386,5504,21930,87552,349866,1398784,5593770,22372352,
%T A192466 89483946,357924864,1431677610,5726666752,22906579626,91626143744,
%U A192466 366504225450,1466016202752,5864063412906,23456250855424,93824997829290
%N A192466 Coefficient of x in the reduction by x^2->x+2 of the polynomial p(n,x)=1+x^n+x^(2n).
%C A192466 For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
%H A192466 Yuanan Diao, Michael Finney, and Dawn Ray, <a href="https://arxiv.org/abs/2007.02819">The number of oriented rational links with a given deficiency number</a>, arXiv:2007.02819 [math.GT], 2020. See p. 16.
%F A192466 Empirical G.f.: -2*x*(x^2 - 3*x + 1) / ((x - 1)*(x + 1)*(2*x - 1)*(4*x - 1)). - _Colin Barker_, Nov 12 2012
%F A192466 Conjectures from _Colin Barker_, Feb 14 2017: (Start)
%F A192466 a(n) = (-1 - (-1)^n + 2^n + 4^n) / 3.
%F A192466 a(n) = 6*a(n-1) - 7*a(n-2) - 6*a(n-3) + 8*a(n-4) for n>4.
%F A192466 (End)
%e A192466 The first four polynomials p(n,x) and their reductions are as follows:
%e A192466 p(1,x)=1+x+x^2 -> 3+2x
%e A192466 p(2,x)=1+x^2+x^4 -> 9+6x
%e A192466 p(3,x)=1+x^3+x^6 -> 25+24x
%e A192466 p(4,x)=1+x^4+x^8 -> 93+90x.
%e A192466 From these, read
%e A192466 A192465=(3,9,25,93,...) and A192466=(2,6,24,90,...)
%t A192466 (See A192465.)
%Y A192466 Cf. A192232, A192465, A192467.
%K A192466 nonn
%O A192466 1,1
%A A192466 _Clark Kimberling_, Jul 01 2011