Squares, triangles, and octal
by John from John D. Cook on (#6RJ1G)
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.
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