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A192430 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments. 2
1, 1, 3, 9, 33, 113, 403, 1409, 4977, 17489, 61619, 216809, 763377, 2686881, 9458787, 33295297, 117206177, 412579681, 1452347043, 5112464521, 17996645761, 63350804881, 223004208243, 785007489729, 2763341973393, 9727369663793 (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

Conjecture: a(n) = 2*a(n-1)+6*a(n-2)-2*a(n-3)-a(n-4). G.f.: -(x^3+5*x^2+x-1) / (x^4+2*x^3-6*x^2-2*x+1). - Colin Barker, May 12 2014

EXAMPLE

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

p(0,x)=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

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

u[x_] := x + d; v[x_] := x - d;

p[n_, x_] := (u[x]^n + v[x]^n)/2 + (u[x]^n - v[x]^n)/(2 d) (* A163762 *)

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}] (* A192430 *)

Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192431 *)

CROSSREFS

Cf. A192232, A192431.

Sequence in context: A037129 A148987 A176812 * A323790 A148988 A148989

Adjacent sequences:  A192427 A192428 A192429 * A192431 A192432 A192433

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jun 30 2011

STATUS

approved

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Last modified November 29 04:38 EST 2021. Contains 349416 sequences. (Running on oeis4.)