On certain ratios regarding integer numbers which are both triangulars and squares

Abstract: We investigate integer numbers which possess at the same time the properties to be triangulars and squares, that are, numbers $a$ for which do exist integers $m$ and $n$ such that $ a = n^2 = \frac{m \cdot (m+1)}{2} $. In particular, we are interested about ratios between successive numbers of that kind. While the limit of the ratio for increasing $a$ is already known in literature, to the best of our knowledge the limit of the ratio of differences of successive ratios, again for increasing $a$, is a new investigation. We give a result for the latter limit, showing that it coincides with the former one, and we formulate a conjecture about related limits.