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[parent] Cahen's constant (Definition)

Whereas a simple addition of unit fractions with the terms of Sylvester's sequence as denominators gives as a result the integer 1, an alternating sum

$\displaystyle \sum_{i = 0}^\infty \frac{(-1)^i}{a_i - 1}$
(where $ a_i$ is the $ i$th term of Sylvester's sequence) gives the transcendental number known as Cahen's constant (after Eugène Cahen) with an approximate decimal value of 0.643410546288338026182254307757564763286587860268239505987 (see A118227 in Sloane's OEIS). Alternatively, we can express Cahen's constant as
$\displaystyle \sum_{j = 0}^\infty \frac{1}{a_{2j}}.$
The recurrence relation $ b_{n + 2} = {b_n}^2b_{n + 1} + b_n$ gives us the terms for the continued fraction representation of this constant:
$\displaystyle 1 + \frac{1}{{b_0}^2 + \frac{1}{{b_1}^2 + \frac{1}{{b_3}^2 + \, \cdots}}}$



"Cahen's constant" is owned by Mravinci.
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Other names:  Cahen constant

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Cross-references: representation, continued fraction, recurrence relation, OEIS, transcendental number, alternating sum, integer, denominators, Sylvester's sequence, terms, unit fractions, addition, simple
There are 2 references to this entry.

This is version 2 of Cahen's constant, born on 2006-11-12, modified 2006-11-12.
Object id is 8549, canonical name is CahensConstant.
Accessed 1050 times total.

Classification:
AMS MSC: 11A55 (Number theory :: Elementary number theory :: Continued fractions)
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Continued fraction problem continues... by Mravinci on 2006-11-12 16:00:06
Continued fraction looks good in "page images" but not so good in "HTML with images." There's at least one other, older, PM entry with this problem.
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