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

[israel at math.ubc.ca]

For n = 8, I can prove there is no possible m. If m had d digits, and the 
concatenation was term j of the sequence (with 8 as term 0 and m as term 
1), we would have

10^d*8 + m = 8*F(j-1) + m*F(j)

thus

m = 8*(10^d - F(j-1))/(F(j)-1)

We'd need m >= 10^(d-1),  which says 

(8/(F(j)-1) - 1/10) 10^d - 8 F(j-1)/(F(j)-1) >= 0

The left side would be negative if F(j) >= 81. Thus we need j <= 10. On the 
other hand, m < 10^d, and it's easily seen this is impossible if j <= 5. 
For j = 6, m = (8*10^d - 40)/7 < 10^d requires 10^d < 40, so we'd need d=1, 
but that doesn't work since (8*10-40)/7 is not an integer. For j = 7, 
2*10^d-16 == 1 mod 3 so m = (2*10^d - 16)/3 is never an integer. For j = 8, 
2*10^d-26 == 4 mod 5 so m = (2*10^d - 26)/5 is never an integer. For j = 9, 
8*10^d-168 = 2 mod 3 so m = (8*10^d - 168)/33 is never an integer. For j = 
10, 4*10^d-136 is never divisible by 27 so m = (4*10^d-136)/27 is never an 
integer.

Similar considerations should work in the other cases.

For comparison, if you take n=5, the constraint m >= 10^(d-1) says
j <= 9, while m < 10^d is impossible if j <= 4.  For j from 5 to 9, the
only one that works is j = 6: 
m = (5*10^d - 25)/7 is an integer for d == 5 mod 6, giving the solutions
m = (5*10^d - 25)/7 = 71425, 71428571425, 71428571428571425, ...

Cheers,
Robert

On Jan 27 2016, 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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