Lehmer showed that every positive irrational number x has a unique infinite continued cotangent representation of the form x = cot[ sum_(k = 0)^∞ (-1)^k cot^(-1) b_k], where the b_ks are nonnegative and b_k>=b_(k - 1)^2 + b_(k - 1) + 1. Note that this growth condition on coefficients is essential for the uniqueness of Lehmer expansion. The following table summarizes the coefficients b_k for various special constants.

cotangent | inverse cotangent | Lehmer's constant

Derrick Henry Lehmer