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A192426 Constant term of the reduction by x^2->x+2 of the polynomial p(n,x) defined below in Comments. 2
2, 0, 5, 1, 18, 13, 81, 106, 413, 729, 2258, 4653, 12833, 28666, 74493, 173545, 437346, 1041421, 2583089, 6221322, 15304541, 37079289, 90826994, 220729069, 539487297, 1313161498, 3205831869, 7809748489, 19054635650, 46439068365 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The polynomial p(n,x) is defined by ((x+d)/2)^n+((x-d)/2)^n, where d=sqrt(x^2+8).  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..29.

FORMULA

Conjecture: a(n)=a(n-1)+5*a(n-2)-2*a(n-3)-4*a(n-4). G.f.: -(5*x^2+2*x-2) / (4*x^4+2*x^3-5*x^2-x+1). - Colin Barker, May 12 2014

EXAMPLE

The first five polynomials p(n,x) and their reductions are as follows:

p(0,x)=2 -> 2

p(1,x)=x -> x

p(2,x)=4+x^2 -> 5+x

p(3,x)=6x+x^3 -> 1+8x

p(4,x)=8+8x^2+x^4 -> 18+11x.

From these, read A192426=(2,0,5,1,18,...) and A192427=(0,1,1,8,11,...)

MATHEMATICA

q[x_] := x + 1; d = Sqrt[x^2 + 8];

p[n_, x_] := ((x + d)/2)^n + ((x - d)/2)^n (* suggested by A162514 *)

Table[Expand[p[n, x]], {n, 0, 6}]

reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};

t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 0, 30}]

Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192426 *)

Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192427 *)

CROSSREFS

Cf. A192232, A192427.

Sequence in context: A290395 A082974 A167635 * A075603 A264357 A221573

Adjacent sequences:  A192423 A192424 A192425 * A192427 A192428 A192429

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jun 30 2011

STATUS

approved

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Last modified September 21 23:10 EDT 2020. Contains 337274 sequences. (Running on oeis4.)