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 A041019 #49 Aug 10 2024 21:38:22
%S A041019 1,1,2,3,5,33,38,71,109,180,1189,1369,2558,3927,6485,42837,49322,
%T A041019 92159,141481,233640,1543321,1776961,3320282,5097243,8417525,55602393,
%U A041019 64019918,119622311,183642229,303264540,2003229469,2306494009,4309723478,6616217487,10925940965
%N A041019 Denominators of continued fraction convergents to sqrt(13).
%H A041019 Vincenzo Librandi, <a href="/A041019/b041019.txt">Table of n, a(n) for n = 0..200</a>
%H A041019 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,36,0,0,0,0,1).
%F A041019 From _Johannes W. Meijer_, Jun 12 2010: (Start)
%F A041019 a(5*n) = A006190(3*n+1),
%F A041019 a(5*n+1) = (A006190(3*n+2) - A006190(3*n+1))/2,
%F A041019 a(5*n+2) = (A006190(3*n+2) + A006190(3*n+1))/2,
%F A041019 a(5*n+3) = A006190(3*n+2) and a(5*n+4) = A006190(3*n+3)/2. (End)
%F A041019 G.f.: ((1 - 2*x + 4*x^2 - 3*x^3 + x^4)*(1 + 3*x + 4*x^2 + 2*x^3 + x^4))/(1 - 36*x^5 - x^10). - _Peter J. C. Moses_, Jul 29 2013
%F A041019 a(n) = A010122(n)*a(n-1) + a(n-2), a(0)=1, a(-1)=0. - _Paul Weisenhorn_, Aug 17 2018
%t A041019 Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[13], n]]], {n, 1, 50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 16 2011 *)
%t A041019 CoefficientList[Series[((1 - 2 x + 4 x^2 - 3 x^3 + x^4) (1 + 3 x + 4 x^2 + 2 x^3 + x^4))/(1 - 36 x^5 - x^10), {x, 0, 30}], x] (* _Vincenzo Librandi_, Dec 10 2013 *)
%t A041019 LinearRecurrence[{0,0,0,0,36,0,0,0,0,1},{1,1,2,3,5,33,38,71,109,180},40] (* _Harvey P. Dale_, Sep 30 2016 *)
%o A041019 (Magma) I:=[1, 1, 2, 3, 5, 33, 38, 71, 109, 180]; [n le 10 select I[n] else 36*Self(n-5)+Self(n-10): n in [1..50]]; // _Vincenzo Librandi_, Dec 10 2013
%Y A041019 Cf. A010122 (continued fraction for sqrt(13)), A041018 (numerators).
%Y A041019 Cf. A041047, A041091, A041151, A041227, A041319, A041427 and A041551. - _Johannes W. Meijer_, Jun 12 2010
%K A041019 nonn,cofr,frac,easy
%O A041019 0,3
%A A041019 _N. J. A. Sloane_
%E A041019 More terms from _Vincenzo Librandi_, Dec 10 2013