Definition:Lehmer's Constant
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Definition
Let the sequence $\sequence {b_k}_{k \mathop \in \N}$ be defined by the recurrence relation:
\(\ds b_k\) | \(=\) | \(\ds \begin {cases} 0 & : k = 0 \\ {b_{k - 1} }^2 + b_{k - 1} + 1 & : \text {otherwise} \end{cases}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sequence {0, 1, 3, 13, 183, 33973 \ldots}\) |
This sequence is A002065 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Lehmer's constant is the real number $\xi$ defined as:
\(\ds \xi\) | \(=\) | \(\ds \map \cot {\sum_{k \mathop = 0}^\infty \paren {-1}^k \arccot b_k}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \cot {\arccot 0 - \arccot 1 + \arccot 3 - \arccot 13 + \arccot 183 - \arccot 33973 + \dotsb}\) | ||||||||||||
\(\ds \) | \(\approx\) | \(\ds 0 \cdotp 59263 \, 27182 \ldots\) |
This sequence is A030125 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Also see
- Definition:Salem Constant (for Raphaël Salem), also seen referred to as Lehmer's constant, otherwise unrelated to this.
Source of Name
This entry was named for Derrick Henry Lehmer.
Sources
- 1938: D.H. Lehmer: A Cotangent Analogue of Continued Fractions (Duke Math. J. Vol. 4: pp. 323 – 340)
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,59263 27182 \ldots$
- Weisstein, Eric W. "Lehmer's Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Lehmer'sConstant.html