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

[Jack Brennen]

Sorry, but there is no m for which the Fibonacci sequence
starting with (8,m) contains the concatenation 8..m.

After you eliminate the small values of m, you are left with
a couple of rounded values for (8..m)/m which are allowable:
13, 21, 34, and 55, meaning that the concatenation 8..m
must be either the 6th, 7th, 8th, or 9th term following
m.

Trying all possible values for the last four digits of m,
only the following are plausible values of m:

m ends with 0828 -> the 6th subsequent term ends with 0828.
m ends with 0904 -> the 8th subsequent term ends with 0904.
m ends with 3328 -> the 6th subsequent term ends with 3328.
m ends with 5828 -> the 6th subsequent term ends with 5828.
m ends with 8328 -> the 6th subsequent term ends with 8328.

(All other values of m fail to match the last four digits
at any of the 6th, 7th, 8th, or 9th terms.)

Assume that the part of m excluding the last four digits
is A.  A = floor(m/10000).

Then we have five possibilities to exclude:

    8,10000*A+828,  ..., 130000*A+10828
    8,10000*A+904,  ..., 340000*A+30904
    8,10000*A+3328, ..., 130000*A+43328
    8,10000*A+5828, ..., 130000*A+75828
    8,10000*A+8328, ..., 130000*A+108328

The differences between m and the candidate concatenation
are respectively:

    120000*A+10000
    330000*A+30000
    120000*A+40000
    120000*A+70000
    120000*A+100000

One of those would need to be equal to 8 times a power
of 10.  That's impossible, since 8 times a power of 10
is always 2 (mod 3), and none of those values are ever
2 (mod 3).

- Jack



On 1/27/2016 7:29 AM, Eric Angelini wrote:
> 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.
>
>
>
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