Davison, J. L.; Shallit, J. O. Continued fractions for some alternating series. (English) Zbl 0719.11038 Monatsh. Math. 111, No. 2, 119-126 (1991). We discuss certain simple continued fractions that exhibit a type of “self-similar” structure: their partial quotients are formed by perturbing and shifting the denominators of their convergents. We prove that all such continued fractions represent transcendental numbers. As an application, we prove that Cahen’s constant \(C=\sum_{i\geq 0}\frac{(- 1)^ i}{S_ i-1}\) is transcendental. Here \((S_ n)\) is Sylvester’s sequence, defined by \(S_ 0=2\) and \(S_{n+1}=S^ 2_ n-S_ n+1\) for \(n\geq 0\). We also explicitly compute the continued fraction for the number C; its partial quotients grow doubly exponentially and they are all squares. Reviewer: J.L.Davison Cited in 1 ReviewCited in 12 Documents MSC: 11J70 Continued fractions and generalizations 11J82 Measures of irrationality and of transcendence Keywords:simple continued fractions; transcendental numbers; Cahen’s constant; Sylvester’s sequence PDFBibTeX XMLCite \textit{J. L. Davison} and \textit{J. O. Shallit}, Monatsh. Math. 111, No. 2, 119--126 (1991; Zbl 0719.11038) Full Text: DOI EuDML Online Encyclopedia of Integer Sequences: a(n) = (a(n-1) + 1)*a(n-2). Denominators of convergents to Cahen’s constant: a(n+2) = a(n)^2*a(n+1) + a(n). Partial quotients in continued fraction expansion of Cahen’s constant. Partial quotients in continued fraction expansion of 2C-1, where C is Cahen’s constant. a(n+2) = (a(n) - 1)*a(n+1) + 1. Decimal expansion of Cahen’s constant. Decimal expansion of a constant associated with self-generating continued fractions and Cahen’s constant. References: [1] Adams, W. W., Davison, J. L.: A remarkable class of continued fractions. Proc. 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