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

0, 1, 13, 196, 257087, 249639161983, 553029809670900697241813, 575598315149214535162520163688459972096324096213, 680813056961507163626080261194823226597566577785481001106845521689287461487322891517719568410606

Offset: 0

Author

Benoit Cloitre, Apr 10 2003

Keywords

• nonn

References

• Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 433-434.
• D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340.

Links

• Steven R. Finch, Lehmer's Constant [broken link]
• Steven R. Finch, Lehmer's Constant [From the Wayback machine]

Crossrefs

Cf. A002065, A002666, A002667, A002668.

Programs

PARI

\p900
bn=vector(100);
bn[1]=sqrt(3)/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(3)/2 = cot(Sum_{n>=0} (-1)^n*acot(a(n))).

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)).