A192346 -id:A192346 - cp's OEIS frontend

A192347 Coefficient of x in the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

0, 1, 2, 11, 32, 125, 418, 1511, 5248, 18601, 65250, 230099, 809248, 2849989, 10030018, 35311375, 124293632, 437545489, 1540200002, 5421774299, 19085364000, 67183428301, 236495292002, 832498651511, 2930516834432, 10315851565625

Offset: 1

Clark Kimberling, Jun 28 2011

nonn

To define the polynomials p(n,x), let d=sqrt(x+2); 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.

			The first four polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=x -> x
p(2,x)=2+x+x^2 -> 3+2x
p(3,x)=6x+3x^2+x^3 -> 4+11x.
From these, we read
A192346=(1,0,3,4,...) and A192347=(1,1,2,11...)

Cf. A192232, A192346.

Mathematica
```
(See A192346.)
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Conjecture: a(n) = 2*a(n-1)+6*a(n-2)-2*a(n-3)-a(n-4). G.f.: x^2*(x^2+1) / (x^4+2*x^3-6*x^2-2*x+1). [Colin Barker, Jan 17 2013]

cp's OEIS Frontend

A192347 Coefficient of x in the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

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