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

1,3

Comments

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.

Links

Table of n, a(n) for n=1..26.

Formula

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]

Example

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...)

Mathematica

(See A192346.)

Crossrefs

Cf. A192232, A192346.

Sequence in context: A094792 A173707 A332612 * A031400 A085786 A034128

Adjacent sequences: A192344 A192345 A192346 * A192348 A192349 A192350

Keyword

nonn

Author

Clark Kimberling, Jun 28 2011

Status

approved