A081787 Continued cotangent for sqrt(e).

1, 4, 208, 51198, 3265038057, 25300257957809599598, 1548008157389016603196793951803038609594, 15445738611564165990406534887324277271178568836676520360367688416251534382546319

Offset: 0

Author

Benoit Cloitre, Apr 10 2003

Keywords

• nonn

References

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

Crossrefs

Cf. A002666, A002667, A002668.

Programs

PARI

\p900
bn=vector(100);
bn[1]=exp(1/2);
b(n)=if(n<0,0,bn[n]);
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));

Formula

sqrt(e) = cot(Sum_{n>=0} (-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)).