[seqfan] Re: Fibonacci concatenated (a list and some patterns)

[Eric Angelini]

Hello SeqFans,
Jean-Marc Falcoz has computed around 50 terms of a nice seq.
Here is the idea.
Let "n" be the first term of a Fibonacci-like seq
and "m" the smallest integer such that the concatenation "nm"
is part of the said Fibo-like seq. The first 10 terms are:

n m

1 4
2 8
3 23
4 7
5 71425
6 1
7 5
8 0
9 11
10 0

Explanation #1:
14 is part of the Fibo-like 1,4,5,9,14
28 is part of the Fibo-like 2,8,10,18,28
323 is part of the Fibo-like 3,23,26,49,75,124,199,323
47 is part of the Fibo-like 4,7,11,18,29,47
571425 is part of ...
The last couple [5,71425] means that no other m < 71425
produces a concatenation "nm" with n that is part of its
own Fibo-like seq.

Explanation #2:
The "0" that follows 8 doesn't mean that the concatenation
"80" is part of its own Fibo-like seq -- but means that 
no "m" has been found < 10^7 [both Jean-Marc and me are
quite sure that _all_ possible couple "nm" will be part
at some point of its own Fibo-like seq. -- look at the
entry 56 of the hereunder list, for instance, where almost
one and a half million integers have been discarded before 
the first "hit"].

More comments after the list:

n m

1 4
2 8
3 23
4 7
5 71425
6 1
7 5
8 0
9 11
10 0
11 0
12 2
13 0
14 0
15 445
16 0
17 0
18 3
19 0
20 87
21 623
22 0 
23 0
24 4
25 0
26 1802
27 33
28 0
29 107
30 5
31 0
32 0
33 79
34 0
35 0
36 6
37 0
38 0
39 2703
40 1063805
41 0
42 7
43 0
44 0
45 805
46 0
47 0
48 8
49 0
50 0
51 0
52 3604
53 0
54 9
55 0
56 1489327
57 0
58 214
59 0
60 0
61 0
62 0
63 77   
64 1702088
...

Jean-Marc wanted to test the gaps between two "hits" 
produced by the same "n" but a different "m". He found
quite a number of astonishing patterns. Here is what 
he gets for n=1 to n=30:

n <--> various m that produce a hit

1 <--> 4, 9, 49, 99, 499, 999, 4999, 9999, 14285, 49999, 
       99999, 499999, 999999, 4999999, etc.

2 <--> 8, 98, 998, 9998, 28570, 99998, 999998, 9999998, etc.

3 <--> 23, 248, 2498, 42855, 249998, 2499998, etc.

4 <--> 7, 97, 997, 9997, 57140, 99997, 999997, 9999997, etc.

5 <--> 71425, ?

6 <--> 1, 46, 178, 496, 4996, 18178, 49996, 85710, 499996, 
       1818178, 4999996, etc.

7 <--> 5, 95, 995, 9995, 99995, 999995, 9999995, etc.

8 <--> ?

9 <--> 11, 69, 161, 267, 744, 1661, 7494, 16661, 27267, 74994,
       166661, 749994, 1666661, 2727267, 7499994, etc.

10 <--> ?
11 <--> ?

12 <--> 2, 92, 356, 992, 9992, 36356, 99992, 999992, 3636356,
        9999992, etc.

13 <--> ?
14 <--> ?

15 <--> 445, 45445, 4545445, etc.

16 <--> ?
17 <--> ?

18 <--> 3, 22, 322, 534, 3322, 33322, 54534, 333322, 3333322,
        5454534, etc.

19 <--> ?

20 <--> 87, 987, 9987, 99987, 999987, 9999987, etc.

21 <--> 623, 63623, 6363623, etc.

22 <--> ?
23 <--> ?

24 <--> 4, 712, 72712, 7272712, etc.

23 <--> ?

26 <--> 1802, 181802, etc.

27 <--> 33, 483, 801, 4983, 49983, 81801, 499983, 4999983,
        8181801

28 <--> ?

29 <--> 107, 1232, 12482, 124982, 1249982, etc.

30 <--> 5, 890, 90890, 9090890, etc.

...

Jean-Marc tells at 16:28 (Belgian and Swiss time) that the
search for a "m" producing a hit with "8" has reached 10^9
without success...

Best,
É.