A188599 Expansion of x/(1-6*x+25*x^2).

0, 1, 6, 11, -84, -779, -2574, 4031, 88536, 430441, 369246, -8545549, -60504444, -149387939, 616283466, 7432399271, 29187308976, -10686127919, -793799491914, -4495643753509, -7128875223204, 69617842498501, 595928935571106, 1835127550964111, -3887458083492984

Offset

0,3

Comments

Original name: G.f.: 1/(1-6*x+25*x^2).

Suggested by Philippe Flajolet as an example of a simple formula for which the general term is hard to guess because 1-6*x+25*x^2 has 2 complex roots of equal size and modulus 1.

The Lucas sequence U_n(6,25). - Peter Bala, Feb 02 2017

References

Discussion in 1993 at the FPSAC 1993 in Florence.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 110 terms from Vincenzo Librandi)

Wikipedia, Lucas sequence

Index entries for linear recurrences with constant coefficients, signature (6,-25).

Formula

a(n) = ((3+4*i)^n-(3-4*i)^n)/8/i, where i=sqrt(-1). - Denis Excoffier, Jan 19 2013

From Peter Bala, Feb 02 2017: (Start)

a(n) = (1/4)*( Re((2 - i)^n)*Im((2 + i)^n) - Re((2 + i)^n)*Im((2 - i)^n) ).

a(n) = (1/2) * the directed or signed area of the triangle in the complex plane with vertices at the points 0, (2 - i)^n and (2 + i)^n. (End)

a(n) = 5^n*sin(n*arctan(1/2))*cos(n*arctan(1/2))/2. - Peter Luschny, Feb 02 2017

E.g.f.: (1/4)*exp(3*x)*sin(4*x). - Stefano Spezia, Feb 01 2020

Maple

x/(1-6*x+25*x^2):series(%, x, 44):seriestolist(%);

Mathematica

Table[Im[(3 + 4*I)^n]/4, {n, 0, 22}] (* Jean-François Alcover, Jun 14 2011 *)
CoefficientList[Series[x/(1-6*x+25*x^2), {x, 0, 30}], x] (* Harvey P. Dale, Dec 01 2018 *)
LinearRecurrence[{6, -25}, {0, 1}, 30] (* Harvey P. Dale, Jul 03 2021 *)

Prog

(PARI) Vec(x/(1-6*x+25*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jun 14 2011

Crossrefs

Sequence in context: A219702 A013321 A192893 * A217934 A012419 A012663

Adjacent sequences: A188596 A188597 A188598 * A188600 A188601 A188602

Keyword

sign,easy

Author

Simon Plouffe, Apr 06 2011

Extensions

Minor edits by N. J. A. Sloane, Apr 06 2011

Minor modification to Name by Peter Bala, Feb 02 2017

Status

approved