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A192357 Constant term of the reduction of the polynomial p(n,x)=(1/2)((x+3)^n+(x-3)^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 (list; graph; refs; listen; history; text; internal format)
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(n-1)+19*a(n-2)-20*a(n-3)-55*a(n-4). G.f.: x*(x^3-9*x^2-2*x+1)/((5*x^2+5*x+1)*(11*x^2-7*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

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Last modified December 7 20:04 EST 2021. Contains 349585 sequences. (Running on oeis4.)