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.svg) |
(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.
 | OEIS |  |
| 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.svg) |
(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.svg) |
(4)
|
where 
is a Fibonacci number (B. Cloitre, pers.
comm., Sep. 22, 2005).
