

A192357


Constant term of the reduction of the polynomial p(n,x)=(1/2)((x+3)^n+(x3)^n) by x^2>x+1.


2



1, 0, 10, 1, 137, 93, 2219, 3410, 39586, 94467, 750823, 2317249, 14833565, 53482716, 301162922, 1194377453, 6225350029, 26179063845, 130188268471, 567580989502, 2742763551458, 12225952022559, 58052436966875, 262325736910601
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OFFSET

1,3


COMMENTS

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=1..24.


FORMULA

Conjecture: a(n) = 2*a(n1)+19*a(n2)20*a(n3)55*a(n4). G.f.: x*(x^39*x^22*x+1)/((5*x^2+5*x+1)*(11*x^27*x+1)). [Colin Barker, Nov 22 2012]


MATHEMATICA

q[x_] := x + 1; d = 3;
p[n_, x_] := ((x + d)^n + (x  d)^n )/2 (* similar to polynomials defined at A161516 *)
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[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
(* A192357 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A192358 *)


CROSSREFS

Cf. A192232, A192358.
Sequence in context: A287753 A185544 A048882 * A156286 A049223 A308282
Adjacent sequences: A192354 A192355 A192356 * A192358 A192359 A192360


KEYWORD

nonn


AUTHOR

Clark Kimberling, Jun 29 2011


STATUS

approved



