The idea can be generalized as (u, v)-Ulam numbers by selecting different starting values (u, v). A sequence of (u, v)-Ulam numbers is regular if the sequence of differences between consecutive numbers in the sequence is eventually periodic. When v is an odd number greater than three, the (2, v)-Ulam numbers are regular. When v is congruent to 1 (mod 4) and at least five, the (4, v)-Ulam numbers are again regular. However, the Ulam numbers themselves do not appear to be regular. first observed the regularity of the sequences for u = 2 and v = 7 and v = 9. conjectured the extension of this result to all odd v greater than three, and this conjecture was proven by . The regularity of the (4, v)-Ulam numbers was proven by .
