OFFSET
0,2
COMMENTS
Together with A002320 these are the two sequences satisfying ( a(n)^2+a(n-1)^2 )/(1 - a(n)a(n-1)) is an integer, in both cases this integer is -5. - Floor van Lamoen, Oct 26 2001
Limit_{n->infinity} a(n+1)/a(n) = (5 + sqrt(21))/2 = A107905. - Wolfdieter Lang, Nov 17 2023
REFERENCES
From a posting to Netnews group sci.math by ksbrown(AT)seanet.com (K. S. Brown) on Aug 15 1996.
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Margherita Maria Ferrari and Norma Zagaglia Salvi, Aperiodic Compositions and Classical Integer Sequences, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.8.
Tanya Khovanova, Recursive Sequences
MathPages, N = (x^2 + y^2)/(1+xy) is a Square
Index entries for linear recurrences with constant coefficients, signature (5,-1).
FORMULA
Sequences A002310, A002320 and A049685 have this in common: each one satisfies a(n+1) = (a(n)^2+5)/a(n-1). - Graeme McRae, Jan 30 2005
G.f.: (1-3x)/(1-5x+x^2). - Philippe Deléham, Nov 16 2008
a(n) = S(n, 5) - 3*S(n-1, 5), for n >= 0, with the S-Chebyshev polynomial (see A049310) S(n, 5) = A004254(n+1). - Wolfdieter Lang, Nov 17 2023
MATHEMATICA
LinearRecurrence[{5, -1}, {1, 2}, 25] (* T. D. Noe, Feb 22 2014 *)
PROG
(Haskell)
a002310 n = a002310_list !! n
a002310_list = 1 : 2 :
(zipWith (-) (map (* 5) (tail a002310_list)) a002310_list)
-- Reinhard Zumkeller, Oct 16 2011
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Joe Keane (jgk(AT)jgk.org)
STATUS
approved