An observation on A091407

[Geoffrey Caveney]

Sequence A091407 lists the values of the lengths of the "y" blocks for the first 98 terms of Gijswijt's sequence (A090822), where the sequence up to a(n-1) is expressed in the form xy^k with the maximal value of k.

I report the surprising observation that every value of the length of y will occur--with the unique exception of 2 ! The values 1 and 3 occur early, and the progression of the higher level-m sequences (A091787, A091799, A091844, etc.) quickly reveals that each has an early repeated y-block with length m+2, such as 2223 2223..., 33334 33334..., 444445 444445..., etc. This ensures the occurrence of every length of y >= 3 in A091407 if it is extended for a sufficient number of terms.

But a repeated y block with length 2 is not to be found in Gijswijt's sequence. In particular, the strings 212, 323, 434, etc., are not found, which prevents the occurrence of 1212, 2121, 2323, 3232, etc. In the case of 212, it can be proven not to occur within any string of 1's and 2's in the sequence by working backward and reaching a contradiction: each of the strings 2212, 221212, 121212, 111212, 2211212, 21211212, and 11211212 violates the sequence, and these comprise all possible strings of 1's and 2's ending in 212.

I suspect that it is likely more difficult to prove that 323 cannot occur in any string of 2's and 3's in Gijswijt's sequence. The non-occurrence of 323 is relevant to the unproven Curling Number Conjecture (Finiteness Conjecture), since all of the longest tail strings following the most stubborn starting strings of 2's and 3's in the analysis of the Conjecture contain numerous occurrences of the string 323.

Geoffrey Caveney