A002310 a(n) = 5*a(n-1) - a(n-2), with a(0) = 1 and a(1) = 2.

1, 2, 9, 43, 206, 987, 4729, 22658, 108561, 520147, 2492174, 11940723, 57211441, 274116482, 1313370969, 6292738363, 30150320846, 144458865867, 692144008489, 3316261176578, 15889161874401, 76129548195427, 364758579102734, 1747663347318243, 8373558157488481, 40120127440124162

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A002310, A002320, A004254, A049310, A049685, A054477, A107905.

Programs

Haskell

a002310 n = a002310_list !! n
a002310_list = 1 : 2 :
   (zipWith (-) (map (* 5) (tail a002310_list)) a002310_list)
-- Reinhard Zumkeller, Oct 16 2011

Mathematica

LinearRecurrence[{5, -1}, {1, 2}, 25] (* T. D. Noe, Feb 22 2014 *)

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