Cahen's constant

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In mathematics, Cahen's constant is defined as an infinite series of unit fractions, with alternating signs, derived from Sylvester's sequence:

C = \sum\frac{(-1)^i}{s_i-1}=\frac11 - \frac12 + \frac16 - \frac1{42} + \frac1{1806} - \cdots\approx 0.64341054629.

By considering these fractions in pairs, we can also view Cahen's constant as a series of positive unit fractions formed from the terms in even positions of Sylvester's sequence; this series for Cahen's constant forms its greedy Egyptian expansion:

C = \sum\frac{1}{s_{2i}}=\frac12+\frac17+\frac1{1807}+\frac1{10650056950807}+\cdots

This constant is named after Eugène Cahen (also known for the Cahen-Mellin integral), who first formulated and investigated its series (Cahen 1891).

Cahen's constant is known to be transcendental (Davison & Shallit 1991). It is notable as being one of a small number of naturally occurring transcendental numbers for which we know the complete continued fraction expansion: if we form the sequence

1, 1, 2, 3, 14, 129, 25298, 420984147, ... (sequence A006279 in OEIS)

defined by the recurrence relation

q_{n+2} = q_n^2 q_{n+1} + q_n

then the continued fraction expansion of Cahen's constant is

[0,1,q_0^2,q_1^2,q_2^2,\ldots]

(Davison & Shallit 1991).

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