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On the average of triangular numbers

Mario Catalani
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Abstract

The problem we are dealing with is the following: find two sequences $a_n$ and $b_n$ such that the average of the first $b_n$ triangular numbers (starting with the triangular number 1) is still a triangular number, precisely the $a_n$-th triangular number. We get also some side results: for instance one of the sequence instrumental to finding the asked for sequences turns out to be a bisection of the sequence of the numerators of continued fraction convergents to $\sqrt{3}$.

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Mario Catalani. 2003-04-13. On the average of triangular numbers. https://arxiv.org/abs/math/0304160

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