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

Author

Simon Plouffe, Apr 06 2011

Keywords

• sign
• easy

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).

Programs

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 *)

PARI

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

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

Extensions

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

Minor modification to Name by Peter Bala, Feb 02 2017