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A192431 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

%I A192431 #8 Jun 07 2019 22:01:55
%S A192431 0,1,4,15,52,185,648,2287,8040,28321,99660,350879,1235036,4347705,
%T A192431 15304208,53873695,189642192,667570433,2349942420,8272149359,
%U A192431 29119170180,102503781241,360828342424,1270168882575,4471181087032,15739215003425
%N A192431 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
%C A192431 The polynomial p(n,x) is defined by (u^n+v^n)//2)^n+(u^n-v^n)/(2d), where u=x+d, v=x-d, d=sqrt(x^2+2).  For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.
%F A192431 Conjectures from _Colin Barker_, Jun 07 2019: (Start)
%F A192431 G.f.: x*(1 + x)^2 / (1 - 2*x - 6*x^2 + 2*x^3 + x^4).
%F A192431 a(n) = 2*a(n-1) + 6*a(n-2) - 2*a(n-3) - a(n-4) for n>3.
%F A192431 (End)
%e A192431 The first five polynomials p(n,x) and their reductions are as follows:
%e A192431 p(0,x)=1 -> 1
%e A192431 p(1,x)=1+x -> 1+x
%e A192431 p(2,x)=2+3x+x^2 -> 3+4x
%e A192431 p(3,x)=2+7x+6x^2+x^3 -> 9+15x
%e A192431 p(4,x)=4+12x+17x^2+10x^3+x^4 -> 33+52x.
%e A192431 From these, read
%e A192431 A192430=(1,1,3,9,33,...) and A192431=(0,1,4,15,52,...)
%t A192431 (See A192430.)
%Y A192431 Cf. A192232, A192430.
%K A192431 nonn
%O A192431 0,3
%A A192431 _Clark Kimberling_, Jun 30 2011