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

Neil Sloane njasloane at gmail.com
Sat Jul 1 01:20:03 CEST 2017

```Kerry, Can you please submit your tribonacci version?
It should certainly be in the OEIS.
(And add a cross-reference to it in A104324 so
people can find it)

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Email: njasloane at gmail.com

On Fri, Jun 30, 2017 at 6:57 PM, Kerry Mitchell <lkmitch at gmail.com> wrote:

> 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/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

```