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Date: 12-10-2018
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Date: 25-4-2019
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Date: 30-3-2019
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Lehmer (1938) showed that every positive irrational number has a unique infinite continued cotangent representation of the form
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(1) |
where the s are nonnegative and
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(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.
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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, ... |
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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
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(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
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(4) |
where is a Fibonacci number (B. Cloitre, pers. comm., Sep. 22, 2005).
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."
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