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A192349 Coefficient of x in the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

%I A192349 #7 Jan 17 2013 09:05:12
%S A192349 0,1,2,14,40,180,616,2456,8960,34384,128160,485728,1823360,6882368,
%T A192349 25896064,97614720,367575040,1384954112,5216465408,19651804672,
%U A192349 74025216000,278859191296,1050447030272,3957059508224,14906157629440,56151566438400
%N A192349 Coefficient of x in the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.
%C A192349 To define the polynomials p(n,x), let d=sqrt(x+3); then p(n,x)=(1/2)((x+d)^n+(x-d)^n).  These are similar to polynomials at A161516. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
%F A192349 Conjecture: a(n) = 2*a(n-1)+8*a(n-2)-4*a(n-3)-4*a(n-4). G.f.: x^2*(2*x^2+1) / (4*x^4+4*x^3-8*x^2-2*x+1). [_Colin Barker_, Jan 17 2013]
%e A192349 The first four polynomials p(n,x) and their reductions are as follows:
%e A192349 p(0,x)=1 -> 1
%e A192349 p(1,x)=x -> x
%e A192349 p(2,x)=3+x+x^2 -> 4+2x
%e A192349 p(3,x)=9x+3x^2+x^3 -> 4+14x.
%e A192349 From these, we read
%e A192349 A192348=(1,0,3,4,...) and A192349=(0,1,2,14...)
%t A192349 (See A192348.)
%Y A192349 Cf. A192232, A192348.
%K A192349 nonn
%O A192349 1,3
%A A192349 _Clark Kimberling_, Jun 28 2011