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

0,3

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.

Links

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

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)

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)=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...)

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.)

Crossrefs

Cf. A192232, A192350.

Sequence in context: A139271 A133217 A001504 * A136905 A230644 A183907

Adjacent sequences: A192348 A192349 A192350 * A192352 A192353 A192354

Keyword

nonn

Author

Clark Kimberling, Jun 28 2011

Extensions

Offset corrected by Robert Israel, Jan 01 2018

Status

approved