Unexpected behaviour of crossing sequences

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.