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a sequence that becomes self-referencing

Source: Brazilian Undergrad Mathematics Olympiad 2022 P3

November 26, 2022
number theoryreal analysis

Problem Statement

Let (an)n∈N(a_n)_{n \in \mathbb{N}} be a sequence of integers. Define an(0)=ana_n^{(0)} = a_n for all n∈Nn \in \mathbb{N}. For all M≥0M \geq 0, we define (an(M+1))n∈N: an(M+1)=an+1(M)−an(M),∀n∈N(a_n^{(M + 1)})_{n \in \mathbb{N}}:\, a_n^{(M + 1)} = a_{n + 1}^{(M)} - a_n^{(M)}, \forall n \in \mathbb{N}. We say that (an)n∈N(a_n)_{n \in \mathbb{N}} is (M + 1)-self-referencing\textrm{(M + 1)-self-referencing} if there exists k1k_1 and k2k_2 fixed positive integers such that an+k1=an+k2(M+1),∀n∈Na_{n + k_1} = a_{n + k_2}^{(M + 1)}, \forall n \in \mathbb{N}.
(a) Does there exist a sequence of integers such that the smallest MM such that it is M-self-referencing\textrm{M-self-referencing} is M=2022M = 2022?
(a) Does there exist a stricly positive sequence of integers such that the smallest MM such that it is M-self-referencing\textrm{M-self-referencing} is M=2022M = 2022?
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