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# Lehmer Cotangent Expansion

Lehmer (1938) showed that every positive irrational number has a unique infinite continued cotangent representation of the form

 (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

 (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

 (4)

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

Cotangent, Inverse Cotangent, Lehmer's Constant

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## References

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."

## Referenced on Wolfram|Alpha

Lehmer Cotangent Expansion

## Cite this as:

Weisstein, Eric W. "Lehmer Cotangent Expansion." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LehmerCotangentExpansion.html