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Continued cotangent for sqrt(e).
0

%I #11 Jul 11 2015 01:05:49

%S 1,4,208,51198,3265038057,25300257957809599598,

%T 1548008157389016603196793951803038609594,

%U 15445738611564165990406534887324277271178568836676520360367688416251534382546319

%N Continued cotangent for sqrt(e).

%D D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340.

%F sqrt(e)=cot(sum(n>=0, n, (-1)^n*acot(a(n))); let b(0)=sqrt(e), b(n)=(b(n-1)*floor(b(n-1))+1)/(b(n-1)-floor(b(n-1)) then a(n)=floor(b(n)).

%o (PARI) ?bn=vector(100); b(n)=if(n<0,0,bn[n]); bn[1]=sqrt(e); ?for(n=2,10,bn[n]=(b(n-1)*floor(b(n-1))+1)/(b(n-1)-floor(b(n-1)))) ?a(n)=floor(b(n+1))

%Y Cf. A002666, A002667, A002668.

%K nonn

%O 0,2

%A _Benoit Cloitre_, Apr 10 2003