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A081796 Continued cotangent for sin(Pi/3) = sqrt(3)/2.

%I A081796 #19 Apr 24 2025 06:12:43
%S A081796 0,1,13,196,257087,249639161983,553029809670900697241813,
%T A081796 575598315149214535162520163688459972096324096213,
%U A081796 680813056961507163626080261194823226597566577785481001106845521689287461487322891517719568410606
%N A081796 Continued cotangent for sin(Pi/3) = sqrt(3)/2.
%D A081796 Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 433-434.
%D A081796 D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340.
%H A081796 Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/iteration/article/">Lehmer's Constant</a> [broken link]
%H A081796 Steven R. Finch, <a href="http://web.archive.org/web/20010603070928/http://www.mathsoft.com/asolve/constant/lehmer/lehmer.html">Lehmer's Constant</a> [From the Wayback machine]
%F A081796 sqrt(3)/2 = cot(Sum_{n>=0} (-1)^n*acot(a(n))).
%F A081796 Let b(0) = sqrt(3)/2, b(n) = (b(n-1)*floor(b(n-1))+1)/(b(n-1)-floor(b(n-1))) then a(n) = floor(b(n)).
%o A081796 (PARI) \p900
%o A081796 bn=vector(100);
%o A081796 bn[1]=sqrt(3)/2;
%o A081796 b(n)=if(n<0,0,bn[n]);
%o A081796 for(n=2,10,bn[n]=(b(n-1)*floor(b(n-1))+1)/(b(n-1)-floor(b(n-1))));
%o A081796 a(n)=floor(b(n+1));
%Y A081796 Cf. A002065, A002666, A002667, A002668.
%K A081796 nonn
%O A081796 0,3
%A A081796 _Benoit Cloitre_, Apr 10 2003