[seqfan] Re: Post's tag system, continued

[Don Reble]

> Don, if there were an infinite orbit starting at length <= 46, would
> your program have found it?

    Yes, and then it would follow that infinite loop.
    Update 1: I find that there is no infinite orbit starting at
      length <= 57.
    Update 2: Ach! there are errors in sequences A289670,1,6,7
      starting at A(43). Sorry about that. (I send corrections to njas.)

> Assuming the answer is Yes, then we know that
> A289670(n) + A289671(n) = 2^n (because for n <= 46, every word either
> dies or goes into a cycle). Assuming that is true, I've extended
> A289671 from your b-file for A289671.

    Good.

> Second question: If we number the bits in the word w from left to right
> starting at bit 0, then as Don pointed out, the bits 1,2,4,5,7,8,.,,,
> (not a multiple of 3) are "junk DNA" and have no affect on the outcome
> of w under the Post tag system.
>
> This means that we can divide the counts in [A289670] and A289671 by
> 2 to the power f1(n), where f1(n) is the number of non-multiples of 3
> in the range 0 to n-1. That is, f1(n) = 2*floor((n-1)/3)+((n+2) mod 3).
> Don, do you agree?

    Yes.

> The reduced counts ... are now A289676 and A289677


-- 
Don Reble  djr at nk.ca