The existence of (p,q)-extended Rosa sequences

A (p,qp,q)-extended Rosa sequence is a sequence of length 2n+22n+2 containing each of the symbols 0,1,…,n0,1,…,n exactly twice, and such that two occurrences of the integer j>0j>0 are separated by exactly j-1j-1 symbols. We prove that, with two exceptions, the conditions necessary for the existence of a (p,qp,q)-extended Rosa sequence with prescribed positions of the symbols 0 are sufficient. We also extend the result to λλ-fold (p,qp,q)-extended Rosa sequences; i.e., the sequences where every pair of numbers is repeated exactly λλ times.