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Divisibility of a certain sequence

Source: Germany 2014-Problem 2

December 4, 2022
combinatoricsnumber theoryDivisibilitySequenceformula

Problem Statement

For a positive integer nn, let yny_n be the number of nn-digit positive integers containing only the digits 2,3,5,72,3,5, 7 and which do not have a 55 directly to the right of a 2.2. If r≥1r\geq 1 and m≥2m\geq 2 are integers, prove that ym−1y_{m-1} divides yrm−1.y_{rm-1}.
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