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K-pop sequences

Source: India EGMO TST 2023/5

December 10, 2022
combinatorics

Problem Statement

Let kk be a positive integer. A sequence of integers a1,a2,⋯a_1, a_2, \cdots is called kk-pop if the following holds: for every n∈Nn \in \mathbb{N}, ana_n is equal to the number of distinct elements in the set {a1,⋯ ,an+k}\{a_1, \cdots , a_{n+k} \}. Determine, as a function of kk, how many kk-pop sequences there are.
Proposed by Sutanay Bhattacharya
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