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 A123251 #27 Oct 06 2023 10:29:45 %S A123251 1,4,1,3,1,12,1,7,1,20,1,11,1,28,1,15,1,36,1,19,1,44,1,23,1,52,1,27,1, %T A123251 60,1,31,1,68,1,35,1,76,1,39,1,84,1,43,1,92,1,47,1,100,1,51,1,108,1, %U A123251 55,1,116,1,59,1,124,1,63,1,132,1,67,1,140,1,71,1,148,1,75,1,156,1,79,1 %N A123251 Continued fraction for sqrt(2)*tan(1/sqrt(2)). %C A123251 This continued fraction allows us to see that tan(1/sqrt(2)), sin(1/sqrt(2)), cos(1/sqrt(2)) are irrational. More generally, for any fixed positive integer m, the continued fraction for sqrt(m)*tan(1/sqrt(m)) is given by a(12*n-11) = a(12*n-9) = a(12*n-7) = a(12*n-5) = a(12*n-3) = a(12*n-1) = 1; a(12*n-10) = 12*m*n - 9*m - 2; a(12*n-8) = 12*n-9; a(12*n-6) = 12*m*n - 5*m - 2; a(12*n-4) = 12*n-5; a(12*n-2) = 12*m*n - m - 2; a(12*n) = 12*n-1. %C A123251 From _Peter Bala_, Oct 02 2023: (Start) %C A123251 Further to the above, the simple continued fraction expansion for sqrt(2)*tan(sqrt(2)/2) may be derived by setting z = sqrt(2)/2 in Lambert's continued fraction tan(z) = z/(1 - z^2/(3 - z^2/(5 - ... ))) and, after using an equivalence transformation, making repeated use of the identity 1/(n - 1/m) = 1/((n - 1) + 1/(1 + 1/(m - 1))) together with further equivalence transformations. %C A123251 The same approach can be used to find the simple continued fraction expansions for the numbers sqrt(N)*tan(sqrt(N)/(k*N)) and (sqrt(N)/N)*tan(sqrt(N)/(k*N)) for k >= 1. An example is given below. (End) %H A123251 G. C. Greubel, <a href="/A123251/b123251.txt">Table of n, a(n) for n = 1..1000</a> %F A123251 For n >= 1 we have a(12*n-11) = a(12*n-9) = a(12*n-7) = a(12*n-5) = a(12*n-3) = a(12*n-1) = 1; a(12*n-10) = 24*n-20; a(12*n-8) = 12*n-9; a(12*n-6) = 24*n-12; a(12*n-4) = 12*n-5; a(12*n-2) = 24*n-4; a(12*n) = 12*n-1. %F A123251 Empirical g.f.: x*(x^7 - x^6 + 4*x^5 - x^4 + 3*x^3 + x^2 + 4*x + 1) / ((x-1)^2*(x+1)^2*(x^2+1)^2). - _Colin Barker_, Jun 28 2013 %F A123251 a(2*n-1) = 1, a(4*n) = 4*n-1 and a(4*n-2) = 8*n-4 for n >= 1. - _Peter Bala_, Oct 02 2023 %e A123251 From _Peter Bala_ Oct 03 2023: (Start) %e A123251 For k > 1, the simple continued fraction expansion of sqrt(2)*tan(sqrt(2)/(2*k)) is [0; k - 1, 1, 2*3*k - 2, 1, 5*k - 2, 1, 2*7*k - 2, 1, 9*k - 2, 1, 2*11*k - 2, 1, 13*k - 2, 1, 2*15*k - 2, 1, ...], and the simple continued fraction expansion of (sqrt(2)/2)*tan(sqrt(2)/(2*k)) is [0; 2*k - 1, 1, 3*k - 2, 1, 2*5*k - 2, 1, 7*k - 2, 1, 2*9*k - 2, 1, 11*k - 2, 1, 2*13*k - 2, 1, 15*k - 2, 1, ...]. (End) %p A123251 cfrac(sqrt(2)*tan(1/sqrt(2)),81,'quotients'); # _Muniru A Asiru_, Oct 13 2018 %t A123251 ContinuedFraction[Sqrt[2]*Tan[1/Sqrt[2]], 100] (* _G. C. Greubel_, Oct 12 2018 *) %o A123251 (PARI) contfrac(sqrt(2)*tan(1/sqrt(2))) \\ _G. C. Greubel_, Oct 12 2018 %o A123251 (Magma) continuedFraction(Sqrt(2)*Tan(1/Sqrt(2))); // _G. C. Greubel_, Oct 12 2018 %Y A123251 Cf. A123168. %K A123251 nonn,cofr %O A123251 1,2 %A A123251 _Benoit Cloitre_, Oct 08 2006