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

$$\sum _{i=0}^{\mathrm{\infty }}\frac{{(-1)}^i}{a_i-1}$$

(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

$$\sum _{j=0}^{\mathrm{\infty }}\frac{1}{a_{2j}}.$$

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:

$$1+\frac{1}{b_0{}^2+\frac{1}{b_1{}^2+\frac{1}{b_3{}^2+\mathrm{\cdots }}}}$$