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

0,3

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.

Links

Table of n, a(n) for n=0..25.

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)

Example

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,...)

Mathematica

(See A192430.)

Crossrefs

Cf. A192232, A192430.

Sequence in context: A132894 A117917 A369671 * A329253 A161125 A027295

Adjacent sequences: A192428 A192429 A192430 * A192432 A192433 A192434

Keyword

nonn

Author

Clark Kimberling, Jun 30 2011

Status

approved