Kevin McCall, whose name you may recognize because his thoughts on dice and the Tarot (qv) have appeared on this blog in the past, has worked out a proof of the congruence patterns in the series of triangular numbers which I postulated here. He has proven that the series of triangular numbers reduced modulo k repeats itself, that the repetition has a period of k (if k is odd) or 2k (if k is even), and that the repeating series is always palindromic.
I have only seen his proof of the first of those statements; I am not going to look at the remainder of his proof until after I have proven it myself independently -- at which point I will post both his proof and mine.
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