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

M. F. Hasler seqfan at hasler.fr
Sat Jul 1 19:27:02 CEST 2017

```This can be generalized to the "k-bonacci" limit for any k >= 2,
with x-> (x, ..., x+k-1) if k|x, and x -> x+1 else.
(For k=1 the initial [0] would be a fixed point and thus equal to the
limit).
The matrix starting with row 2 looks as follows:
[0 1 2 2 3 2 3 4 2 3  4 4 5 2 3 4 4 5  4  5  6  2  3  4  4]
[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]
[0 1 2 3 2 3 4 3 4 4  5 6 7 4 4 5 6 7  4  5  6  7  6  7  8]
[0 1 2 3 4 2 3 4 5 3  4 5 5 6 7 8 9 4  5  5  6  7  8  9  5]
[0 1 2 3 4 5 2 3 4 5  6 3 4 5 6 6 7 8  9 10 11  4  5  6  6]
[0 1 2 3 4 5 6 2 3 4  5 6 7 3 4 5 6 7  7  8  9 10 11 12 13]
[0 1 2 3 4 5 6 7 2 3  4 5 6 7 8 3 4 5  6  7  8  8  9 10 11]
[0 1 2 3 4 5 6 7 8 2  3 4 5 6 7 8 9 3  4  5  6  7  8  9  9]
Read by falling antidiagonals this is
0,1,0,2,1,0,2,2,1,0,3,2,2,1,0,2,3,3,2,1,0,3,3,2,3,2,1,0,4,3,
3,4,3,2,1,0,2,4,4,2,4,3,2,1,0,3,5,3,3,5,4,3,2,1,0,...
Tentatively submitted as oeis.org/draft/A289281 <https://oeis.org/A289281>.

- Maximilian

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.
> ...
>

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

```