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->oo} 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
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 44, 56.
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
E.g.f.: exp(5*x/2)*(21*cosh(sqrt(21)*x/2) - sqrt(21)*sinh(sqrt(21)*x/2))/21. - Stefano Spezia, Jul 07 2025
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
AUTHOR
Joe Keane (jgk(AT)jgk.org)
STATUS
approved