Summary: In an unpublished 2005 paper [Polynômes de type Legendre et approximations de la constante d’Euler, Unpublished preprint (2005), http://www-fourier.ujf-grenoble.fr/~rivoal/articles/euler.pdf] T. Rivoal proved the formula \[ \frac 4\pi= \prod_{k \geq 2}\left( 1 + \frac 1 {k +1}\right)^{2\rho(k) \lfloor\log_2(k)-1\rfloor} \] where \(\lfloor x\rfloor\) denotes the (lower) integer part of the real number \(x\), and \(\rho(k)\) is the 4-periodic sequence defined by \(\rho(0) = 1\), \(\rho(1) =-1\), \(\rho(2)= \rho(3) = 0\). We show how a lemma in a 1988 paper of J. Shallit and the author [Analytic number theory, Proc. Jap.-Fr. Symp., Tokyo/Jap. 1988, Lect. Notes Math. 1434, 19–30 (1990; Zbl 0711.11003)] allows us to prove that formula, as well as a family of similar formulas involving occurrences of blocks of digits in the base-\(B\) expansion of the integer \(k\), where \(B\) is an integer \(\geq 2\).