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A192351 Coefficient of x in the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments. 2
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A139271 A133217 A001504 * A136905 A230644 A183907
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jun 28 2011
EXTENSIONS
Offset corrected by Robert Israel, Jan 01 2018
STATUS
approved

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