This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A002310 #72 Jul 10 2025 04:32:36
%S A002310 1,2,9,43,206,987,4729,22658,108561,520147,2492174,11940723,57211441,
%T A002310 274116482,1313370969,6292738363,30150320846,144458865867,
%U A002310 692144008489,3316261176578,15889161874401,76129548195427,364758579102734,1747663347318243,8373558157488481,40120127440124162
%N A002310 a(n) = 5*a(n-1) - a(n-2), with a(0) = 1 and a(1) = 2.
%C A002310 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
%C A002310 Limit_{n->oo} a(n+1)/a(n) = (5 + sqrt(21))/2 = A107905. - _Wolfdieter Lang_, Nov 17 2023
%D A002310 From a posting to Netnews group sci.math by ksbrown(AT)seanet.com (K. S. Brown) on Aug 15 1996.
%H A002310 Reinhard Zumkeller, <a href="/A002310/b002310.txt">Table of n, a(n) for n = 0..1000</a>
%H A002310 Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, <a href="http://dx.doi.org/10.13140/RG.2.2.16511.73129">Lune and Lens Sequences</a>, ResearchGate preprint, 2024. See pp. 44, 56.
%H A002310 Margherita Maria Ferrari and Norma Zagaglia Salvi, <a href="https://www.emis.de/journals/JIS/VOL20/Salvi/salvi3.html">Aperiodic Compositions and Classical Integer Sequences</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.8.
%H A002310 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A002310 MathPages, <a href="http://www.mathpages.com/home/kmath334.htm">N = (x^2 + y^2)/(1+xy) is a Square</a>
%H A002310 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-1).
%F A002310 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
%F A002310 G.f.: (1-3x)/(1-5x+x^2). - _Philippe Deléham_, Nov 16 2008
%F A002310 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
%F A002310 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
%t A002310 LinearRecurrence[{5, -1}, {1, 2}, 25] (* _T. D. Noe_, Feb 22 2014 *)
%o A002310 (Haskell)
%o A002310 a002310 n = a002310_list !! n
%o A002310 a002310_list = 1 : 2 :
%o A002310 (zipWith (-) (map (* 5) (tail a002310_list)) a002310_list)
%o A002310 -- _Reinhard Zumkeller_, Oct 16 2011
%Y A002310 Cf. A002310, A002320, A004254, A049310, A049685, A054477, A107905.
%K A002310 nonn,easy
%O A002310 0,2
%A A002310 Joe Keane (jgk(AT)jgk.org)