A123336 Values X satisfying the equation 3(X-Y)^4-16XY=0, where X>=Y.

0, 3, 32, 405, 5488, 75867, 1054560, 14680173, 204438752, 2847353715, 39658107808, 552364642437, 7693441239120, 107155791629643, 1492487562920672, 20787669795714525, 289534888481556928

Offset

0,2

Comments

Sequence gives X values. To find Y values: b(n) = c(n)*(-1 + d(n)) which gives: 0, 1, 24, 375, 5376, 75449, 1053000, 14674351, 204417024, 2847272625, 39657805176,...

Links

G. C. Greubel, Table of n, a(n) for n = 0..870

Index entries for linear recurrences with constant coefficients, signature (18,-58,18,-1).

Formula

a(n) = c(n)*(1 + d(n)) with c(0)=0,c(1)=1 and c(n) = 4*c(n-1) - c(n-2) d(0)=1,d(1)=2 and d(n) = 4*d(n-1) - d(n-2).

From Max Alekseyev, Nov 13 2009: (Start)

For n>=4, a(n) = 18*a(n-1) - 58*a(n-2) + 18*a(n-3) - a(n-4).

O.g.f: x*(3*x^2 - 22*x + 3)/((x^2-14*x+1)*(x^2-4*x+1)). (End)

a(n) = (((2 + sqrt(3))^n - (2 - sqrt(3))^n)*(2 + (2 - sqrt(3))^n + (2 + sqrt(3))^n))/(4*sqrt(3)). - Gerry Martens, Jun 06 2015

Mathematica

CoefficientList[Series[x*(3*x^2 - 22*x + 3)/(x^2 - 14*x + 1)/(x^2 - 4*x + 1), {x, 0, 50}], x] (* G. C. Greubel, Oct 12 2017 *)
LinearRecurrence[{18, -58, 18, -1}, {0, 3, 32, 405}, 20] (* Harvey P. Dale, Feb 11 2018 *)

Prog

(PARI) x='x+O('x^50); concat([0], Vec(x*(3*x^2 - 22*x + 3)/((x^2-14*x+1)*(x^2-4*x+1)))) \\ G. C. Greubel, Oct 12 2017
(Magma) [Round((((2 + Sqrt(3))^n - (2 - Sqrt(3))^n)*(2 + (2 - Sqrt(3))^n + (2 + Sqrt(3))^n))/(4*Sqrt(3))): n in [0..30]]; // G. C. Greubel, Oct 12 2017

Crossrefs

Sequence in context: A137215 A300366 A231591 * A058479 A264334 A278069

Adjacent sequences: A123333 A123334 A123335 * A123337 A123338 A123339

Keyword

nonn

Author

Mohamed Bouhamida, Oct 11 2006

Extensions

More terms from Max Alekseyev, Nov 13 2009

Status

approved