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On Moessner’s Theorem

Dexter Kozen and Alexandra Silva
The American Mathematical Monthly
Vol. 120, No. 2 (February 2013), pp. 131-139
DOI: 10.4169/amer.math.monthly.120.02.131
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Page Count: 9
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On Moessner’s Theorem
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Abstract Moessner’s theorem describes a procedure for generating a sequence of n integer sequences that lead unexpectedly to the sequence of nth powers 1n, 2n, 3n, … Paasche’s theorem is a generalization of Moessner’s; by varying the parameters of the procedure, we can obtain the sequence of factorials 1!, 2!, 3!, … or the sequence of superfactorials 1!, 2! 1!, 3! 2! 1!, … Long’s theorem generalizes Moessner’s in another direction, providing a procedure to generate the sequence a · 1n−1, (a + d) · 2n−1, (a + 2d) · 3n−1, …. Proofs of these results in the literature are typically based on combinatorics of binomial coefficients or calculational scans. In this note, we give a short and revealing algebraic proof of a general theorem that contains Moessner’s, Paasche’s, and Long’s as special cases. We also prove a generalization that gives new Moessner-type theorems.

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