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

0, 1, 4, 15, 52, 185, 648, 2287, 8040, 28321, 99660, 350879, 1235036, 4347705, 15304208, 53873695, 189642192, 667570433, 2349942420, 8272149359, 29119170180, 102503781241, 360828342424, 1270168882575, 4471181087032, 15739215003425

Offset: 0

Author

Clark Kimberling, Jun 30 2011

Keywords

• nonn

Comments

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.

Examples

			The first five polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=1+x -> 1+x
p(2,x)=2+3x+x^2 -> 3+4x
p(3,x)=2+7x+6x^2+x^3 -> 9+15x
p(4,x)=4+12x+17x^2+10x^3+x^4 -> 33+52x.
From these, read
A192430=(1,1,3,9,33,...) and A192431=(0,1,4,15,52,...)
		

Crossrefs

Cf. A192232, A192430.

Programs

Mathematica

(See A192430.)

Formula

Conjectures from Colin Barker, Jun 07 2019: (Start)

G.f.: x*(1 + x)^2 / (1 - 2*x - 6*x^2 + 2*x^3 + x^4).

a(n) = 2*a(n-1) + 6*a(n-2) - 2*a(n-3) - a(n-4) for n>3.

(End)