A002668 Continued cotangent for e.

2, 8, 75, 8949, 119646723, 15849841722437093, 708657580163382065836292133774995, 529026553215766321676623343348414600292754204772300344704877695232

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

References

• D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340.
• 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 = 1..11
• D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340. [Annotated scanned copy]
• Eric Weisstein's World of Mathematics, Lehmer Cotangent Expansion

Programs

Mathematica

b[1] = E; b[n_] := b[n] = (b[n-1]*Floor[b[n-1]]+1) / (b[n-1]-Floor[b[n-1]]); a[n_] := Floor[b[n]]; Table[a[n], {n, 1, 8}] (* Jean-François Alcover, Jan 17 2012, after PARI *)

PARI

{ default(realprecision, 10000); bn=vector(11); bn[1]=exp(1); for(n=2, 11, bn[n]=(bn[n-1]*floor(bn[n-1]) + 1)/(bn[n-1] - floor(bn[n-1]))); for (n=1, 11, write("b002668.txt", n, " ", floor(bn[n]))); } \\ Harry J. Smith, May 04 2009

Extensions

More terms from Jeffrey Shallit