[seqfan] Re: The Fibonacci word over the nonneg integers

[David Seal]

As a minor wording point about:

> Trajectory of 0 under the map x -> x,x+1 if x is even, x -> x+1 if x is
> odd, starting at 0:
> 
> 0, 1, 2, 2, 3, 2, 3, 4, 2, 3, 4, 4, 5, 2, 3, ...

I had a bit of trouble understanding what was intended - the map described takes 0 to 0,1, so surely each 'point' on the trajectory contains a 0, so why does the sequence contain only one 0 and not infinitely many?

I have sorted out the answer to that question, but I think I would describe the sequence as something like the limit of the trajectory rather than the trajectory. I.e. the trajectory is the sequence of finite integer sequences:

   0
-> 0,1
-> 0,1,2
-> 0,1,2,2,3
-> 0,1,2,2,3,2,3,4
-> 0,1,2,2,3,2,3,4,2,3,4,4,5
-> 0,1,2,2,3,2,3,4,2,3,4,4,5,2,3,4,4,5,4,5,6
-> ...

and the infinite sequence is in some sense what that trajectory converges to.

Looking at the finite sequences in that trajectory, and in particular how many times each digit appeared in each, was entertaining - it gave me a sort of 'diagonal sums' correspondence between the Fibonacci numbers and Pascal's triangle that was new to me. Probably not actually new - they're very well-studied mathematical constructs! - but quite fun:

                 1 = 1

               1 + 1 = 3
              +
             1   2 + 1 = 8
                +
           1 + 3   3 + 1 = 21
          +       +
         1   4 + 6   4 + 1 = 55
            +       +
       1 + 5   10+10
      +       +
     1   6 +15
        +
   1 + 7
  +
 1

and at the alternate positions:

                 1 = 2
                +
               1   1 = 5
                  +
             1 + 2   1 = 13
            +       +
           1   3 + 3   1 = 34
              +       +
         1 + 4   6 + 4
        +       +
       1   5 +10
          +
     1 + 6
    +
   1

Best regards,

David