Abstract: The discovery of formulas involving mathematical constants such as $\pi$ and $e$ had a great impact on various fields of science and mathematics. However, such discoveries have remained scarce, relying on the intuition of mathematicians such as Ramanujan and Gauss. Recent efforts to automate such discoveries, such as the Ramanujan Machine project, relied solely on exhaustive search and remain limited by the space of options that can be covered. Here we propose a fundamentally different method to search for conjectures on mathematical constants: through analysis of integer sequences. We introduce the Enumerated Signed-continued-fraction Massey Approve (ESMA) algorithm, which builds on the Berlekamp-Massey algorithm to identify patterns in integer sequences that represent mathematical constants. ESMA has found various known formulas and new conjectures for $e, e^2, \tan(1)$, and ratios of values of Bessel functions, many of which provide faster numerical convergence than their corresponding simple continued fractions forms. We also characterize the space of constants that ESMA can catch and quantify its algorithmic advantage in certain scenarios. Altogether, this work continues the development toward algorithm-augmented mathematical intuition, to help accelerate mathematical research.