[seqfan] Re: Fibonaccit

[Maximilian Hasler]

On Fri, Feb 15, 2013 at 6:40 PM, Eric Angelini <Eric.Angelini at kntv.be> wrote:
> Hello SeqFans,
> start S with 0 and 1. Now:
> - the next term of S is the sum of the
>   last two not yet summed digits of S:
>
> S=0,1,1,2,3,5,8,13,9,4,12,13,5,3,3,4,...
> Does S end in a loop?

I did not see a loop in the first 1000 terms but maybe there is a kind of
"chaotic attractor" in the form of a subsequence starting
7,7,8,8,9,9,5,5,1,6,13,14,...
of which an increasingly longer piece of initial terms repeats
infinitely often:
I have put *** at the beginning of that subsequence in the printout below.

Maximilian

0,1,1,2,3,5,8,13,9,4,12,13,5,3,3,4,8,8,6,7,12,16,14,13,8,3,3,7,7,
5,5,4,11,11,6,10,14,12,10,9,5,2,2,2,7,7,1,1,5,5,3,3,1,9,14,7,4,4,9,
14,8,2,6,10,8,6,4,10,10,5,11,11,8,13,10,5,12,10,8,7,1,8,14,10,5,
1,1,1,5,6,2,2,2,9,9,4,4,1,5,6,3,3,1,8,15,8,9,9,5,5,1,5,6,2,2,6,11,8,
4,4,11,18,13,8,5,6,11,9,6,4,9,9,6,13,17,18,14,10,6,6,11,8,4,8,7,2,
9,12,8,5,2,2,9,9,4,11,13,11,7,2,10,15,10,13,18,15,7,4,4,8,8,9,9,5,
5,1,6,12,7,2,9,12,12,15,9,11,10,3,10,13,7,4,11,18,13,5,2,2,4,4,2,
8,9,3,1,1,6,6,1,1,4,4,9,9,6,12,11,8,12,16,17,18,14,10,6,7,7,3,9,9,
11,10,3,3,3,3,6,14,10,2,2,1,3,4,1,1,4,10,11,5,2,2,9,9,4,8,7,4,6,8,
6,10,17,12,4,2,7,12,7,2,5,8,13,18,15,7,3,3,2,9,9,3,3,
/***/ 7,7,8,8,9,9,5,5,1,6,13,14, 10,12,18,10,2,2,1,3,6,6,6,9,7,5,5,1,2,
4,3,4,7,5,2,5,5,1,1,2,6,7,4,11,18,13,12,15,11,10,14,14,7,1,1,8,8,3,6,
6,9,8,3,9,9,7,13,9,4,4,9,9,6,12,10,6,5,11,18,12,6,10,14,15,16,17,18,
14,10,6,7,7,4,4,5,5,1,1,3,3,9,9,1,2,4,3,4,9,12,12,15,16,12,10,6,3,6,7,
7,11,12,7,7,10,6,2,3,8,13,11,5,2,2,9,9,4,4,3,3,6,6,2,2,1,1,5,5,5,11,8,
2,9,16,11,9,12,15,17,11,12,18,16,8,4,12,13,8,13,18,15,7,3,3,1,6,11,6,
2,2,9,9,3,8,7,1,1,5,5,6,6,  /***/ 7,7,8,8,9,9,5,5,1,6,13,14, 11,8,9,10,
6,2,4,6,12,18,10,3,6,7,7,13, 10,3,3,3,3,6,6,7,7, 3,3,1,6,9,9,13,14,8,2,2,
3,9,14,8,1,6,8,5,11,9,4,4,2,6,7,4,11,18,13,8,7,6,9,12,8,4,3,2,6,10,10,6,
2,9,10,11,10,7,7,2,10,10,3,3,6,6,8,8,2,2,3,3,9,9,7,14,12,5,3,3,4,11,9,4,
4,9,9,6,12,10,6,4,7,7,2,7,8,4,11,18,12,11,15,8,2,6,
10,11,12,13,14,15,16,17,18,14,10,6,7,7,4,4,5,5,2,9,17,10,1,6,8,6,10,
7,3,3,9,9,1,3,9,13,14,8,4,4,1,3,6,6,6,9,12,13,14,10,6,4,7,15,18,10,
4,4,5,12,10,4,5,12,10,5,12,9,7,14,13,6,2,10,13,8,6,8,13,11,5,2,2,
9,9,4,11,15,13,15,10,3,10,12,7,5,8,7,1,1,1,6,8,11,10,1,1,2,2,1,7,14,
9,3,1,1,1,3,6,9,12,14,16,10,4,5,6,12,18,16,8,5,5,3,7,8,6,7,5,2,10,13,
8,13,18,15,7,3,3,1,6,10,11,14,9,9,15,12,5,2,2,9,9,3,3,2,2,6,13,10,8,
7,1,1,2,2,3,3,4,4,5,5,6,6, /***/ 7,7,8,8,9,9,5,5,1,6,13,14, 11,8,9,10,
7,11,10,8,8,1,1,7,14,14,7,1,7,10,6,12,18,10,4,12,10,4,4,5,12,12,
8,5,4,9,12,12,15,10,3,3,4,4,5,5,1,6,10,11,8,6,6,9,9,1,4,8,9,6,3,3,
1,4,9,6,3,3,1,5,6,3,11,16,8,5,5,4,9,8,3,1,1,4,11,14,14,9,4,4,2,6,7,
4,11,18,13,5,2,2,6,6,4,4,6,6,1,3,4,1,1,3,9,12,13,15,8,2,2,7,14,9,
2,2,1,1,2,3,4,3,8,8,5,13,12,4,2,2,4,9,15,10,3,3,5,5,7,7,1,4,9,11,
7,3,3,9,9,7,14,13,10,8,10,15,14,13,12,7,3,1,1,4,11,9,4,4,9,9,6,12,
10,6,4,7,7,1,1,2,2,5,13,18,10,6,6,3,7,7,4,11

(PARI)
digonacci(n,d=[0,1])={print1("0,1");for(i=2,n,print1(","a=d[1]+d[2]);
d=concat(vecextract(d,"^1"),digits(a)))}