OEIS A000292: Tetrahedral numbers

A quick tour of tetrahedral numbers (the triangular pyramidal numbers). We cover the formula A_n = binom(n+2, 3) = n(n+1)(n+2)/6, their interpretation as the number of balls in a triangular pyramid, and their role as the sum of the first n triangular numbers. We’ll explore elegant recurrences like A_n = A_{n-2} + n^2, and see how these numbers appear in counting non-decreasing triples, in the Wiener index of a path graph, and in playful links such as the 12 days of Christmas. We’ll also note the rare instances when A_n is a perfect square (n = 1, 2, 48). The takeaway: simple stacking problems can reveal rich structure and surprising connections across mathematics and beyond.

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