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
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