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A192431 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A027853 A132894 A117917 * A329253 A161125 A027295

Adjacent sequences:  A192428 A192429 A192430 * A192432 A192433 A192434

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jun 30 2011

STATUS

approved

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Last modified February 27 03:50 EST 2020. Contains 332299 sequences. (Running on oeis4.)