I ran across the following theorem in Ross Honsberger’s book Mathematical Morsels:
Every odd square ends in 1 in base 8, and if you cut off the 1 you have a triangular number.
A number is an odd square if and only if it is the square of an odd number, so odd squares have the form (2n + 1)².
Both parts of the theorem above follow from the calculation
( (2n + 1)² − 1 ) / 8 = n(n + 1) / 2.
In fact, we can strengthen the theorem. Not only does writing the nth odd square in base 8 and chopping off the final digit give some triangular number, it gives the nth triangular number.