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

Kerry Mitchell lkmitch at gmail.com
Sat Jul 1 00:57:48 CEST 2017

```There's a similar sequence for the Tribonacci word:

Trajectory of 0 under the map x -> x,x+1,x+2 if x mod 3 = 0;
x -> x+1 if x mod 3 = 1; x -> x+1 if x mod 3 = 2.

It begins:

0, 1, 2, 2, 3, 3, 3, 4, 5, 3, 4, 5, 3, 4, 5, 5, 6, 3, 4, 5, 5, 6, 3, 4, 5,
5, 6, 6, 6, 7, 8, etc.

The number of terms in each iteration corresponds to the tribonacci
sequence (beginning with 1, 1, 1):

- 1 term: 0
- 3 terms: 0 1 2
- 5 terms: 0 1 2 2 3
- 9 terms: 0 1 2 2 3 3 3 4 5
- 17 terms: 0 1 2 2 3 3 3 4 5 3 4 5 3 4 5 5 6

When looked at this way, each subsequent iteration is the previous one
copied, then the pre-pre-previous iteration with each term incremented by
3, then the pre-previous iteration, with each term incremented by 3.

Kerry

On Fri, Jun 30, 2017 at 11:06 AM, Neil Sloane <njasloane at gmail.com> wrote:

> Interesting new paper in the Electronic J Combinatorics:
>
> Jiemeng Zhang, Zhixiong Wen, Wen Wu, Some Properties of the Fibonacci
> Sequence on an Infinite Alphabet, Electronic Journal of Combinatorics,
> 24(2) (2017), #P2.52.
>
> http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i2p52/pdf
>
> 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, ...
>
> The sequence was in the OEIS already but with a different definition:
> A104324
>
> It has many interesting properties.
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

```