Lehmer (1938) showed that every positive irrational number  has a unique infinite
continued cotangent representation of
the form
![x=cot[sum_(k=0)^infty(-1)^kcot^(-1)b_k],](/images/equations/LehmerCotangentExpansion/NumberedEquation1.gif) |
(1)
|
where the  s are nonnegative
and
 |
(2)
|
Note that this growth condition on coefficients is essential for the uniqueness of Lehmer expansion.
The following table summarizes the coefficients  for various
special constants.
 | Sloane |  | | e | A002668 | 2, 8, 75, 8949, 119646723, 15849841722437093,
... | Euler-Mascheroni constant  | A081782 | 0, 1, 3, 16, 389, 479403, 590817544217, ... | golden ratio  | A006267 | 1, 4, 76, 439204, 84722519070079276,
... | Lehmer's
constant  | A002065 | 0, 1, 3, 13, 183, 33673, ... |  | A002667 | 3, 73, 8599, 400091364,371853741549033970, ... | Pythagoras's constant  | A002666 | 1, 5, 36, 3406, 14694817,727050997716715,
... |
The expansion for the golden ratio  has the beautiful closed form
![phi=cot[sum_(k=0)^infty(-1)^kcot^(-1)(L_(3^k))],](/images/equations/LehmerCotangentExpansion/NumberedEquation3.gif) |
(3)
|
where  is a Lucas
number.
An illustration of a different cotangent expansion for  that is not
a Lehmer expansion because its coefficients grow too slowly is
![phi=cot[sum_(k=0)^infty(-1)^kcot^(-1)(F_(2k+2))],](/images/equations/LehmerCotangentExpansion/NumberedEquation4.gif) |
(4)
|
where  is a Fibonacci number (B. Cloitre, pers. comm., Sep. 22,
2005).
Lehmer, D. H. "A Cotangent Analogue of Continued Fractions." Duke
Math. J. 4, 323-340, 1938.
Rivoal, T. "Propriétés diophantiennes du développement en cotangente continue de Lehmer." http://www-fourier.ujf-grenoble.fr/~rivoal/articles/cotan.pdf.
Shallit, J. "Predictable Regular Continued Cotangent Expansions." J.
Res. Nat. Bur. Standards Sect. B 80B, 285-290, 1976.
Sloane, N. J. A. Sequences A002065/M2961, A002666/M3983, A002668/M1900, A002667/M3171, A006267/M3699, and A081782 in "The On-Line Encyclopedia of Integer Sequences."
|
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