Observations and Conjectures on Mysterious Sequence (MS)

Over the course of our recent investigations into the behavior of our Turing Machine (TM), we have made some interesting observations regarding a mysterious sequence (MS) of numbers generated by the machine. Here, I’d like to summarize our findings and propose some conjectures that have arisen from our discussions.

1. Observations: 1.1 The MS appears to have a repeating pattern of rules when viewed through run-length encoding. For instance, the sequence 1, 3, 3, 1, 5, 3, 7, 1, 9, 3, 3, 1, 13, 3, 15, 1, 17, 11, 3, 1, 21, 3, 7, 1, 25, 3, 3, 1, 29 reoccurs after some initial terms 14845, 3, 511, 1, 513, 283. 1.2 We have noted that the MS can be mapped to a coded string, where each letter represents a set of TM rules. This has helped us observe repeating “words” or patterns in the data. 1.3 The sequence 2222297 appears to be a recurring “word” or pattern in the TM rule sequence. 1.4 JB has found that the terms ms_{2*i}, for i>0 are equal to ms_p2, where p2 is the greatest power of 2 dividing 2*i.
2. Conjectures: 2.1 Given the repeating patterns observed, we conjecture that there may be an underlying order or structure to the rule application in the TM, potentially linked to the properties of numbers in the sequence. 2.2 The recurring patterns suggest that the TM might be operating in cycles or stages, which could be associated with the particular ‘words’ or rule sequences we have identified. 2.3 The connection between the terms ms_{2*i} and the greatest power of 2 dividing 2*i indicates a possible link between the MS and the properties of binary numbers or operations, which warrants further investigation.

Please note that these observations and conjectures are preliminary, and more rigorous testing and analysis are needed to confirm these patterns and their implications. We look forward to further explorations of this mysterious sequence.