Concatenations using 0 (zero)

Today is self-description time!
Why do things simply when you can complicate?

We have decided that the digit 0 has two meanings:
a) zero
b) concatenate the j-th digit of V to the k-th digit of V.

V = 102, 304, 206, 302, 904, 203, 1302, 603, 204, 104, 207, 902, 403, 506, 105, 607, 2034, 1305, 306, 504, 4102, 901, 2041, 905, 3403, 804, 601, 402, 4104, 209, 4105, ...

The first term of V must be read like this:
102 == "concatenate the 1st digit of V to the 2nd digit of V"
As the first digit of V is 1 and the second is 0, we get the string 10;
The second term of V reads:
304 == "concatenate the 3rd digit of V to the 4th digit of V"
As the third digit of V is 2 and the fourth is 3, we get the string 23;
Now:
206 == "concatenate the 2nd digit of V to the 6th digit of V" == string 04;
302 == "concatenate the 3rd digit of to the 2nd digit of V" == string 20;
904 == "concatenate the 9th digit of to the 4th digit of V" == string 63;
203 == "concatenate the 2nd digit of to the 3rd digit of V" == string 02;
1302 == "concatenate the 13th digit of to the 2nd digit of V" == string 90;
...
The successive such strings (seq W hereunder) spell the original sequence V. Comparison:

= 102, 304, 206, 302, 904, 203, 1302,...
W = 10  23  04  20  63  02  90 ...

Designing V
We wanted V to be the lexicographically earliest sequence of distinct positive terms such that the successive 2-digit concatenations of W reproduce V.

As there are only 0ne hundred distinct 2-digits concatenations, we see that every one of them in must be coded by distinct terms of V.

For instance, the string 04 in W (third term above) comes from the 3rd term of V (which is 206) – but as this 04 shows back later (look at the first picture, second term of W2), we cannot use 206 again there; we will thus code 04 using the integer 506 (and indeed 04 is also the concatenation of the 5th digit of V and the 6th digit of V).

What else can we say? All terms of V will contain exactly one zero – not at the beginning nor at the end of the integer. This is why the first term of V is 102: there would be a contradiction if a(1) = 100 or 101.

Question
Does V stop at some point (for lack of available "next distinct term")?

A nice variant (with additions) here.
Best,
É.








Commentaires

  1. I (and my program) confirm your terms. But why do complicated if you can do simple? Without the 0 it would be at least as nice, IMHO. For example, 12 = concatenate digit #1 with digit #2; 409 = concatenate digit #40 with digit #9. Of course it can be ambiguous (435 = #4 & #35 or #43 and #5?) but it's exactly the same with your 0: 10302 = #103 & # 2 or #1 and #302 ? So the complication has no justification. Without the 0 the sequence would go: 11, 14, 16, 18, 41, 44, 46, 48, 61, 64, 66, 68, 81, 84, 86, 88, 110, 404, 409, 435, 414, 4344, 42, 49, 434, 94, 534, 419, 439, 554, 4443, 99, 141, 443, 4144, 444, 114, 449, 4141, 19, 24, 457, 43, 119, 124, 45... (says my program).

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