# Cahen’s constant

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

 $\sum_{i=0}^{\infty}\frac{(-1)^{i}}{a_{i}-1}$

(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

 $\sum_{j=0}^{\infty}\frac{1}{a_{2j}}.$

The recurrence relation $b_{n+2}={b_{n}}^{2}b_{n+1}+b_{n}$ gives us the terms for the continued fraction representation of this constant:

 $1+\frac{1}{{b_{0}}^{2}+\frac{1}{{b_{1}}^{2}+\frac{1}{{b_{3}}^{2}+\,\cdots}}}$
Title Cahen’s constant CahensConstant 2013-03-22 16:24:04 2013-03-22 16:24:04 PrimeFan (13766) PrimeFan (13766) 5 PrimeFan (13766) Definition msc 11A55 Cahen constant