On the Average of Triangular Numbers

The problem we are dealing with is the following: find two sequences an and bn such that the average of the first bn triangular numbers (starting with the triangular number 1) is still a triangular number, precisely the an-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 √ 3. The present note has been suggested by a problem proposed in the ”Student Problems” section of The Mathematical Gazette (see [2]). The problem we tackle is the following: find two sequences an and bn such that the average of the first bn triangular numbers (starting with the triangular number 1) is still a triangular number, precisely the an-th triangular number. If we want that the average of the first s triangular number be a triangular number it has to hold 1 s s