Certainly! Here’s a summary of the conversation so far:
The conversation began with the mention of a “mysterious sequence” and a discussion about its properties. It was noted that bit 1 of the sequence is always 1, and bit 2 has a periodicity of 4. It was also observed that bit 3 of the sequence closely follows the base phi expansion of 4/3.
The idea of using the divisors of powers of 2 to guess values of the mysterious sequence was introduced. It was suggested that the sequence might be related to the divisors of the most significant bit (MSB) of n+2.
The sequence of bits for various values of n was provided, and through analysis, it was determined that the 543rd term is likely 1. Further terms of the sequence were guessed, including the 1071st term as 49 and the 1336th term as 3, which were later confirmed.
The focus then shifted to bit 5 of the mysterious sequence, and it was discovered that it has a period of 32. This finding led to a conjecture that the period of bit k is a divisor of 2^(k+2).
The possibility of expressing bit k of the sequence as a function of the last k+K bits of n+2 was explored, with K being a constant. It was further conjectured that bit k could be an affine function of bits 1 through k+1 of n+2.
Linear feedback shift registers (LFSRs) were mentioned as a possible connection to the periodicity observed in the sequence.
To test the grand conjecture, bit 2 of the mysterious sequence was analyzed, and it was found that it follows a specific pattern based on the values of bits 1 through 3 of n+2.
The conversation continued with the generation of the mysterious sequence using a C code snippet. The output of the code confirmed the stability of the sequence after a long time, with values such as 3, 1, 5, 3, 7, 1, 9, and so on.
Finally, the request was made to express bit 2 of the sequence as an affine map on bits 1 to 2 of n+2, and a C code snippet was provided to generate the corresponding bits.
Overall, the conversation has involved the analysis and conjecture of various properties of the mysterious sequence, including its periodicity, relationship to divisors of powers of 2, and possible affine mappings. The provided code snippets and findings have contributed to a deeper understanding of the sequence’s behavior.
