A002666 Continued cotangent for square root of 2.

1, 5, 36, 3406, 14694817, 727050997716715, 2074744506784679417243551677046, 46045625970633183674340934067371917846908361831894602280765110

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Harry J. Smith, Table of n, a(n) for n = 0..11
• D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340. [Annotated scanned copy]
• D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340.
• Eric Weisstein's World of Mathematics, Lehmer Cotangent Expansion.

Programs

Mathematica

Floor[NestList[(#*Floor[#]+1)/(#-Floor[#]) &, Sqrt[2], 7]] (* Stefano Spezia, Apr 24 2025 *)

PARI

{ default(realprecision, 10000); bn=vector(12); bn[1]=sqrt(2); for(n=2, 12, bn[n]=(bn[n-1]*floor(bn[n-1]) + 1)/(bn[n-1] - floor(bn[n-1]))); for (n=1, 12, print1(floor(bn[n]), ", ")); } \\ Harry J. Smith, May 04 2009

Extensions

More terms from Jeffrey Shallit