A081795 Continued cotangent for Pi/3.

1, 43, 4975, 87377992, 18385473430682423, 5186411232443302687031694765612941, 47469894147223278266560159220413635233953187522490823346090207081760

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

\p1200
bn=vector(100);
bn[1]=Pi/3;
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

Pi/3 = cot(Sum_{n>=0} (-1)^n*acot(a(n))).

Let b(0) = Pi/3, b(n) = (b(n-1)*floor(b(n-1))+1)/(b(n-1)-floor(b(n-1))) then a(n) = floor(b(n)).