Thanks for this. Another interesting proof is that two consecutive triangular numbers (1, 3, 6, 10, 15…) always sum to a square number (1, 4, 9, 16, 25…). However I think the easiest proof of this is purely visual. The two triangular numbers shove together to make a parallelogram that may be readjusted to show a square in such a way that the generalization becomes intuitive. The Book of Numbers by Conway and Guy, and Gnomon by Midhat Gazale provide lots of information on figurate and polyhedral numbers.

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