Definition:Lehmer's Constant

From ProofWiki
Jump to navigation Jump to search

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


Source of Name

This entry was named for Derrick Henry Lehmer.


Sources