I don't know if this is known (or interesting or significant), but there
are many "pseudo-repetitions" in the Ulam sequence. By
this I mean a relatively large number, n, so that U(s+i+p)=(s+i)+d for
i=0 .. n-1, for some constants s, p, and d. For instance
n=134
d=4131173
The 134 Ulam numbers starting with 51497 (Ulam number, not index) and the
134 Ulam numbers starting at 4182670 each differ by 4131173 (the
corresponding terms, that is).
I've now found such a set of length 251. To state it in another
way, the first differences of these two sections of length 251 are the
same.