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:

$$1+\frac{1}{b_{0}{}^{2}+\frac{1}{b_{1}{}^{2}+\frac{1}{b_{3}{}^{2}+\mathrm{\cdots}}}}$$ 
