A192351 Coefficient of x in the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.

0, 1, 2, 20, 56, 320, 1120, 5312, 20608, 90880, 368640, 1577984, 6522880, 27578368, 114909184, 483328000, 2020573184, 8480555008, 35502817280, 148874461184, 623609118720, 2614000353280, 10952269365248, 45901678641152, 192340840939520

Offset: 0

Author

Clark Kimberling, Jun 28 2011

Keywords

• nonn

Comments

To define the polynomials p(n,x), let d=sqrt(x+5); 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.

Examples

			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)=5+x+x^2 -> 6+2x
p(3,x)=15x+3x^2+x^3 -> 4+20x.
From these, we read
A192350=(1,0,6,4,...) and A192351=(0,1,2,20...)
		

Links

• Robert Israel, Table of n, a(n) for n = 0..1606

Crossrefs

Cf. A192232, A192350.

Programs

Maple

f:= gfun:-rectoproc({a(n) = 2*a(n-1)+12*a(n-2)-8*a(n-3)-16*a(n-4),a(0)=0,a(1)=1,a(2)=2,a(3)=20},a(n),remember):
map(f, [$0..50]); # Robert Israel, Jan 01 2018

Mathematica

(See A192350.)

Formula

Conjecture: a(n) = 2*a(n-1)+12*a(n-2)-8*a(n-3)-16*a(n-4). G.f.: x*(4*x^2+1) / (16*x^4+8*x^3-12*x^2-2*x+1). [Colin Barker, Jan 17 2013]

Confirmation of conjecture by Robert Israel, Jan 01 2018: (Start)

The polynomials p(n,x) have g.f. G(z) = (1-x*z)/(1-2*x*z-5*z^2-x*z^2+x^2*z^2).

The reductions mod x^2-x-1 have g.f. g(z) = (1+x*z-2*z-6*z^2+4*x*z^3)/(1-2*z-12*z^2+8*z^3+16*z^4):

note that the numerator of G(z)-g(z) is divisible by x^2-x-1. (End)

Extensions

Offset corrected by Robert Israel, Jan 01 2018