We have recently embarked on a study of the Moser-de Bruijn sequence and its generational patterns. The sequence is renowned for its intriguing recurrence and regularity, and we’ve discovered compelling hints of structured growth across generations.
Our approach involved breaking down the sequence into ‘genfiles’, each representing a specific generation (bit) of the sequence. Each genfile was obtained by evaluating the status of a specific bit across the entire sequence.
A fascinating observation was made between genfile 10 (g10) and genfile 11 (g11). The last quarter of g11 closely resembled the latter half of g10, with only a minor alteration of replacing the final space with a 1.
This observation led us to conjecture a specific set of rules to generate a new genfile from its predecessor. The rules comprise initiating with a certain number of lines of spaces, followed by appending copies of the previous genfile, then strategically replacing certain ‘1’s with spaces, and lastly changing a specific space into a ‘1’.
However, there’s an element of uncertainty in these rules, particularly in the criterion used for replacing ‘1’s with spaces. This part of the process seems vital for understanding the underlying mechanics of the sequence and its growth patterns, thus warranting further exploration.
It’s crucial to validate these conjectured rules with additional data and generations to ensure the universality of these patterns and not merely a characteristic specific to g10 and g11. If proven, these findings could enrich our understanding of the Moser-de Bruijn sequence and potentially provide insights into related sequences and their properties.
Our ongoing work on this topic aims to further refine these rules, verify them through testing, and uncover the intricate logic woven into the Moser-de Bruijn sequence.
