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 A002320 #50 Jul 12 2025 18:40:38
%S A002320 1,3,14,67,321,1538,7369,35307,169166,810523,3883449,18606722,
%T A002320 89150161,427144083,2046570254,9805707187,46981965681,225104121218,
%U A002320 1078538640409,5167589080827,24759406763726,118629444737803
%N A002320 a(n) = 5*a(n-1) - a(n-2).
%C A002320 Together with A002310 these are the two sequences satisfying the requirement that (a(n)^2 + a(n-1)^2)/(1 - a(n)*a(n-1)) be an integer; in both cases this integer is -5. - _Floor van Lamoen_, Oct 26 2001
%D A002320 From a posting to Netnews group sci.math by ksbrown(AT)seanet.com (K. S. Brown) on Aug 15 1996.
%H A002320 Reinhard Zumkeller, <a href="/A002320/b002320.txt">Table of n, a(n) for n = 0..1000</a>
%H A002320 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 p. 44.
%H A002320 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A002320 MathPages, <a href="http://www.mathpages.com/home/kmath334.htm">N = (x^2 + y^2)/(1+xy) is a Square</a>
%H A002320 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-1).
%F A002320 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 A002320 G.f.: (1-2x)/(1-5x+x^2). - _Philippe Deléham_, Nov 16 2008
%F A002320 a(n) = Sum_{k = 0..n} A238731(n,k)*2^k. - _Philippe Deléham, Mar 05 2014
%F A002320 E.g.f.: exp(5*x/2)*(sqrt(21)*cosh(sqrt(21)*x/2) + sinh(sqrt(21)*x/2))/sqrt(21). - _Stefano Spezia_, Jul 07 2025
%F A002320 From _Peter Bala_, Jul 07 2025: (Start)
%F A002320 a(n) = ( (4 + sqrt(21))*(5 - sqrt(21))^(n+1) - (4 - sqrt(21))*(5 + sqrt(21))^(n+1) )/(2^(n+1)*sqrt(21)).
%F A002320 Sum_{n >= 1} (-1)^(n+1)/(a(2*n) + 5/a(2*n)) = 1/15, since 5/(a(2*n) + 5/a(2*n)) = 1/a(2*n-1) + 1/a(2*n+1).
%F A002320 Sum_{n >= 1} (-1)^(n+1)/(a(2*n-1) + 5/a(2*n-1)) = 1/5, since 5/(a(2*n-1) + 5/a(2*n-1)) = 1/a(2*n-2) + 1/a(2*n). (End)
%t A002320 LinearRecurrence[{5,-1},{1,3},30] (* _Harvey P. Dale_, Nov 13 2014 *)
%o A002320 (Haskell)
%o A002320 a002320 n = a002320_list !! n
%o A002320 a002320_list = 1 : 3 :
%o A002320 (zipWith (-) (map (* 5) (tail a002320_list)) a002320_list)
%o A002320 -- _Reinhard Zumkeller_, Oct 16 2011
%Y A002320 Cf. A054477.
%K A002320 nonn,easy
%O A002320 0,2
%A A002320 Joe Keane (jgk(AT)jgk.org)