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 A083695 #20 Jan 12 2025 14:31:27
%S A083695 0,3,6,27,84,303,1026,3567,12264,42363,146046,503907,1738044,5995623,
%T A083695 20681466,71341047,246089424,848884083,2928215286,10100850987,
%U A083695 34842778404,120189811743,414593515506,1430136089727,4933239756984,17017159962603,58700518710126
%N A083695 a(n) = 2*a(n-1) + 5*a(n-2), with a(0) = 0, a(1) = 3.
%C A083695 A002533(n)/a(n) converges to sqrt(2/3).
%H A083695 Vincenzo Librandi, <a href="/A083695/b083695.txt">Table of n, a(n) for n = 0..1000</a>
%H A083695 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,5).
%F A083695 G.f.: 3x/(1-2x-5x^2).
%F A083695 a(n) = 3[(1+sqrt(6))^n-(1-sqrt(6))^n]/(2*sqrt(6)).
%F A083695 a(n) = 3*A002533(n-1) + a(n-1).
%F A083695 G.f.: 1/Q(0) -1, where Q(k) = 1 + 5*x^2 - (3*k+4)*x + x*(3*k+1 - 5*x)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 07 2013
%t A083695 CoefficientList[Series[3x/(1-2x-5x^2), {x, 0, 25}], x]
%t A083695 LinearRecurrence[{2, 5}, {0, 3}, 27] (* _Arkadiusz Wesolowski_, May 21 2013 *)
%o A083695 (Magma) I:=[0,3]; [n le 2 select I[n] else 2*Self(n-1)+5*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Oct 07 2013
%K A083695 easy,nonn
%O A083695 0,2
%A A083695 Mario Catalani (mario.catalani(AT)unito.it), May 03 2003