Lehmer (1938) showed that every positive irrational number has a unique infinite continued cotangent representation of the form
(1)
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where the s are nonnegative and
(2)
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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
(3)
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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
(4)
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where is a Fibonacci number (B. Cloitre, pers. comm., Sep. 22, 2005).