  Formulas involving fundamental mathematical constants had a great impact on
various fields of science and mathematics, for example aiding in proofs of
irrationality of constants. However, the discovery of such formulas has
historically remained scarce, often perceived as an act of mathematical genius
by great mathematicians such as Ramanujan, Euler, and Gauss. Recent efforts to
automate the discovery of formulas for mathematical constants, such as the
Ramanujan Machine project, relied on exhaustive search. Despite several
successful discoveries, exhaustive search remains limited by the space of
options that can be covered and by the need for vast amounts of computational
resources. 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. The ESMA algorithm
found various known formulas for $e, e^2, tan(1)$, and ratios of values of
Bessel functions. The algorithm further discovered a large number of new
conjectures for these constants, some providing simpler representations and
some providing faster numerical convergence than the corresponding simple
continued fractions. Along with the algorithm, we present mathematical tools
for manipulating continued fractions. These connections enable us to
characterize what space of constants can be found by ESMA and quantify its
algorithmic advantage in certain scenarios. Altogether, this work continues in
the development of augmenting mathematical intuition by computer algorithms, to
help reveal mathematical structures and accelerate mathematical research.
