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FE with interesting statement

Source: 2019 Belarus Team Selection Test 1.1

September 2, 2019
functional equationalgebrafunction

Problem Statement

Does there exist a function f:N→Nf:\mathbb N\to\mathbb N such that f(f(n+1))=f(f(n))+2n−1 f(f(n+1))=f(f(n))+2^{n-1} for any positive integer nn? (As usual, N\mathbb N stands for the set of positive integers.)
(I. Gorodnin)
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