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A192421 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments. 2
2, 0, 3, 1, 8, 8, 28, 43, 111, 204, 466, 924, 2007, 4109, 8740, 18136, 38240, 79799, 167643, 350664, 735554, 1540104, 3228459, 6762553, 14172272, 29691368, 62217172, 130356451, 273144327, 572305140, 1199164498, 2512579140, 5264623167, 11030890949 (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+4).  For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

LINKS

Table of n, a(n) for n=0..33.

FORMULA

Conjecture: a(n) = a(n-1)+3*a(n-2)-a(n-3)-a(n-4). G.f.: -(3*x^2+2*x-2) / (x^4+x^3-3*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)=2+x^2 -> 3+x

p(3,x)=3x+x^3 -> 1+5x

p(4,x)=2+4x^2+x^4 -> 8+7x.

From these, read A192421=(2,0,3,1,8,...) and A192422=(0,1,1,5,7,...)

MATHEMATICA

q[x_] := x + 1; d = Sqrt[x^2 + 4];

p[n_, x_] := ((x + d)/2)^n + ((x - d)/2)^n (* 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}] (* A192421 *)

Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192422 *)

CROSSREFS

Cf. A192232, A192422.

Sequence in context: A241640 A158449 A106533 * A035223 A035184 A257541

Adjacent sequences:  A192418 A192419 A192420 * A192422 A192423 A192424

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jun 30 2011

STATUS

approved

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Last modified September 27 15:29 EDT 2020. Contains 337383 sequences. (Running on oeis4.)