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A192423 Constant term of the reduction by x^2->x+2 of the polynomial p(n,x) defined below in Comments. 3
2, 0, 4, 2, 16, 20, 78, 140, 416, 878, 2324, 5280, 13282, 31200, 76724, 182962, 445376, 1069300, 2591118, 6239980, 15089776, 36389278, 87917284, 212144640, 512334722, 1236606720, 2985883684, 7207831202, 17402424496, 42011258900 (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+2, see A192232.

LINKS

Table of n, a(n) for n=0..29.

FORMULA

Conjecture: a(n) = a(n-1)+4*a(n-2)-a(n-3)-a(n-4). G.f.: -2*(x+1)*(2*x-1) / ((x^2-x-1)*(x^2+2*x-1)). - Colin Barker, May 11 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 -> 4+x

p(3,x)=3x+x^3 -> 2+6x

p(4,x)=2+4x^2+x^4 -> 16+9x.

From these, read A192423=(2,0,4,2,16,...) and A192424=(0,1,1,6,9,...)

MATHEMATICA

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

p[n_, x_] := ((x + d)/2)^n + ((x - d)/2)^n (* A161514 *)

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}]   (* A192423 *)

Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]   (* A192424 *)

Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]   (* A192425 *)

CROSSREFS

Cf. A192232, A192424.

Sequence in context: A244136 A133168 A145382 * A078909 A067458 A088330

Adjacent sequences:  A192420 A192421 A192422 * A192424 A192425 A192426

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jun 30 2011

STATUS

approved

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Last modified December 18 18:44 EST 2014. Contains 252174 sequences.