

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
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OFFSET

0,1


COMMENTS

The polynomial p(n,x) is defined by ((x+d)/2)^n+((xd)/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(n1)+4*a(n2)a(n3)a(n4). G.f.: 2*(x+1)*(2*x1) / ((x^2x1)*(x^2+2*x1)).  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



