# [seqfan] Re: Gilbreath on Fibonacci

Neil Sloane njasloane at gmail.com
Tue Sep 19 10:03:43 CEST 2023

```Sean, very nice example!  Yes, should be easy to prove.  But it is 4am.

Best regards
Neil

Neil J. A. Sloane, Chairman, OEIS Foundation.
Also Visiting Scientist, Math. Dept., Rutgers University,
Email: njasloane at gmail.com

On Mon, Sep 18, 2023 at 10:01 PM Sean A. Irvine <sairvin at gmail.com> wrote:

> Hi,
>
> In terms of getting some traction on the sequences arising from the
> Gilbreath transform, it looks to me like the Gilbreath transform of the
> Fibonacci numbers which is apparently 0,1,1 repeated should be easy to
> prove. In this case, terms with index >= n in row n, are simply the
> Fibonacci numbers themselves and that after each set of three rows we have
> one additional copy of 0,1,1 at the start of the row.  So I think you could
> make this more formal with some kind of induction looking back 3 rows?
>
> 1:  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
> 2:  1, 0, 1, 1, 2, 3, 5,  8, 13, 21, 34, 55, ...
> 3:  1, 1, 0, 1, 1, 2, 3,  5,  8, 13, 21, 34, ...
> 4:  0, 1, 1, 0, 1, 1, 2,  3,  5,  8, 13, 21, ...
> 5:  1, 0, 1, 1, 0, 1, 1,  2,  3,  5,  8, 13, ...
> 6:  1, 1, 0, 1, 1, 0, 1,  1,  2,  3,  5,  8, ...
> 7:  0, 1, 1, 0, 1, 1, 0,  1,  1,  2,  3,  5, ...
> ...
>
> Sean.
>
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>
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