\begin{definition}[Definition:Lehmer's Constant]
Let the sequence $\sequence {b_k}_{k \mathop \in \N}$ be defined by the recurrence relation:
{{begin-eqn}}
{{eqn | l = b_k
      | r = \begin {cases} 0 & : k = 0 \\ {b_{k - 1} }^2 + b_{k - 1} + 1 & : \text {otherwise} \end{cases}
}}
{{eqn | r = \sequence {0, 1, 3, 13, 183, 33973 \ldots}
}}
{{end-eqn}}
{{OEIS|A002065}}
'''Lehmer's constant''' is the real number $\xi$ defined as:
{{begin-eqn}}
{{eqn | l = \xi
      | r = \map \cot {\sum_{k \mathop = 0}^\infty \paren {-1}^k \arccot b_k}
}}
{{eqn | r = \map \cot {\arccot 0 - \arccot 1 + \arccot 3 - \arccot 13 + \arccot 183 - \arccot 33973 + \dotsb}
}}
{{eqn | o = \approx
      | r = 0 \cdotp 59263 \, 27182 \ldots
}}
{{end-eqn}}
{{OEIS|A030125}}
\end{definition}