  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 $\sqrt{3}$.
