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 A019426 #40 Feb 21 2025 13:07:22
%S A019426 0,2,1,7,1,13,1,19,1,25,1,31,1,37,1,43,1,49,1,55,1,61,1,67,1,73,1,79,
%T A019426 1,85,1,91,1,97,1,103,1,109,1,115,1,121,1,127,1,133,1,139,1,145,1,151,
%U A019426 1,157,1,163,1,169,1,175,1,181,1,187,1,193,1,199,1,205,1,211,1,217,1,223,1
%N A019426 Continued fraction for tan(1/3).
%C A019426 The simple continued fraction expansion of 3*tan(1/3) is [1; 25, 1, 3, 1, 61, 1, 7, 1, 97, 1, 11, 1, ..., 36*n + 25, 1, 4*n + 3, 1, ...], while the simple continued fraction expansion of (1/3)*tan(1/3) is [0; 8, 1, 1, 1, 43, 1, 5, 1, 79, 1, 9, 1, 115, 1, 13, 1, ..., 36*n + 7, 1, 4*n + 1, 1, ...]. See my comment in A019425. - _Peter Bala_, Sep 30 2023
%H A019426 Harry J. Smith, <a href="/A019426/b019426.txt">Table of n, a(n) for n = 0..20000</a>
%H A019426 Khalil Ayadi, Chiheb Ben Bechir, and Maher Saadaoui, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Ayadi/ayadi12.html">Continued Fractions with Predictable Patterns and Transcendental Numbers</a>, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.4.
%H A019426 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>
%H A019426 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F A019426 From _Bruno Berselli_, Sep 21 2012: (Start)
%F A019426 G.f.: x*(2+x+3*x^2-x^3+x^4)/(1-x^2)^2.
%F A019426 a(n) = 2*a(n-2)-a(n-4) with n>4, a(0)=0, a(1)=2, a(2)=1, a(3)=7, a(4)=1.
%F A019426 a(n) = 1+3*(1-(-1)^n)*(n-1)/2 with n>1, a(0)=0, a(1)=2.
%F A019426 For k>0: a(2k) = 1, a(4k+1) = 2*a(2k+1)-1 and a(4k+3) = 2*a(2k+1)+5, with a(0)=0, a(1)=2. (End)
%e A019426 0.346253549510575491038543565... = 0 + 1/(2 + 1/(1 + 1/(7 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 13 2009
%t A019426 ContinuedFraction[Tan[1/3], 80] (* _Bruno Berselli_, Sep 21 2012 *)
%o A019426 (PARI) { allocatemem(932245000); default(realprecision, 88000); x=contfrac(tan(1/3)); for (n=0, 20000, write("b019426.txt", n, " ", x[n+1])); } \\ _Harry J. Smith_, Jun 13 2009
%o A019426 (Magma) [n le 1 select 2*n else 1+3*(1-(-1)^n)*(n-1)/2: n in [0..80]]; // _Bruno Berselli_, Sep 21 2012
%Y A019426 Cf. A161012 (decimal expansion of tan(1/3)).
%Y A019426 Cf. continued fractions for tan(1/m): A019425 (m=2), A019427 (m=4), A019428 (m=5), A019429 (m=6), A019430 (m=7), A019431 (m=8), A019432 (m=9), A019433 (m=10), A093178 (m=1).
%K A019426 nonn,easy,cofr
%O A019426 0,2
%A A019426 _David W. Wilson_