On repeated difference sequences of a permutation of N

AbstractLet {ak} be a sequence of positive integers, and let {dk} be its difference sequence with dk = |ak − ak + 1|. Call {ak} a permutation if every positive integer appears exactly once as an ak. Having seen a construction which showed that one could construct a permutation whose difference sequence is also a permutation, Erdös asked if one could construct a permutation for which its difference sequence and all succeeding difference sequences would also be permutations. It is shown here that such a permutation exists