comparisons from sequences related to the " Orloj " clock

Consider these sequences 121212121, 123212321, 1234321234321, 12345432123454321, Can you get all the counting numbers 1, 2, 3, 4, ... from contiguous segments of these sequences ? One of the sequences from the OEIS about the <1234321> example is sequence A028355 It is a kind of problem where intuition seems valid but proving is more difficult It seems <121> and <1234321> do have the surprising property that one can always find connected segments that generate each of the successive counting numbers in turn. This is not true for <12321> and <123454321>