Unexpected behaviour of crossing sequences

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The n-th crossing number of a graph G, denoted crₙ(G), is the minimum number of crossings in a drawing of G on an orientable surface of genus n. We prove that for every a>b>0, there exists a graph G for which cr₀(G) = a, cr₁(G) = b, and cr₂(G) = 0. This provides support for a conjecture of Archdeacon et al. and resolves a problem of Salazar.