Fibonaccit

 

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?

Best,

….

 

____________

 

[Jim Nastos]:

 

Hi Eric,

Can you explain to me why the 4th term is 2?

J.

 

[Eric]:

 

Hi Jim,

 

0 and 1 is 1

1 and 1 is 2

1 and 2 is 3

2 and 3 is 5

3 and 5 is 8

5 and 8 is 13

8 and 1 (the first digit of "13") is 9

1 and 3 (the second digit of "13") is 4

3 and 9 is 12

...

 

[Lars Blomberg]:

 

Hello Eric,

Thank you for a Saturday morning challenge. Well, it was not terribly difficult, but fun!

Extending the series to 10^8+2, I have found no loop.

Here is a histogram of the series:

 

0††† 1

1††† 7745040

2††† 6601748

3††† 5716833

4††† 7690317

5††† 7277147

6††† 7621188

7††† 7081774

8††† 7431603

913251816

107097479

113410444

123583979

133099546

143321937

151706873

161778854

17704563

184878860

 

Note the entries for 9 and 17.

Regards,

Lars

 

[Maximilian Hasler]:

 

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.

 

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)))}

 

[Hans Havermann]:

 

Maximilian Hasler:

> 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,...

 

Here is a slightly different approach to finding quasi-regularity in Ericís sequence. I calculated a large number of terms of the sequence, the positions therein of the number 18 and the first differences of these position numbers. Here is a graph:

 

http://chesswanks.com/num/eighteen.png

 

Those varying-length stretches where the graph drops down to the axis are consecutive ones, representing consecutive eighteens in the original sequence.

 

[Zak Seidov]:

 

I donít know about cycle but s(78532..78532+65) = sixty six 9ís.

 

Conjecture: there may be arbitrary long similar runs.

Zak.

 

 

[Hans Havermann]:

 

Lars Blomberg:

> I have found no loop within the first 10^8 terms.

 

Hey Lars. :)

 

As a non-mathematician I canít quite formalize a proof, but consider those consecutive 18s mentioned in my previous post. A stretch of k 18s will generate roughly 2k 9s in the corresponding summed-digits list, which in turn generate 2k 18s. Thus the number of consecutive 18s will grow without limit and no loops are possible.

 

 

[Franklin T. Adams-Watters]:

 

Thatís a fine proof.

 

[Maximilian Hasler]:

 

FYI, this is now http://oeis.org/A214365.

Any comments & contributions are welcome.

 

__________

 

Any thanks to all contributors Ė case closed!

Best,

….

 

__________

 

P.-S.

 

[…ric, mostly to himself]:

 

Voici quelques nombres N qui, subissant le traitement "Fibonaccit", produisent (se greffent sur) S :

 

N : 11, 12, 23, 35, 58, 813, 139, 394, 9412, 41213, 12135, 21353, 13533, ...

 

S = 0, 1, 1, 2, 3, 5, 8, 13, 9, 4, 12, 13, 5, 3, 3, 4, ...

 

Quelle serait la suite N des nombres qui se greffent sur S ? 2013 fait-il partie de cette suite N ?

 

Sweet is there.