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A362228
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Triangle read by rows: row n is the shortest, then lexicographically earliest sequence of positive integers that takes n iterations of the run transform to reach 1.
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0
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1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1
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OFFSET
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0,2
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COMMENTS
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The run transform replaces each run of consecutive identical values with a single value representing the length of that run.
Note that the shortest sequence is preferred over the lexicographically earliest. For example, for row n=6, we could have had (1,1,2,1,1,2,2), which is lexicographically earlier than (1,2,1,1,2), but the former has 7 terms and the shortest sequence has 5.
It is not sufficient to consider only the integers 1 and 2 inside a program for calculating this sequence, because Samuel B. Reid found a string of length 146 which will transform in 15 iterations of the run transform into 1. While doing so the fourth and fifth iterates of the transform contain a 3. - Thomas Scheuerle, Apr 12 2023
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LINKS
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EXAMPLE
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The triangle begins:
0 1;
1 2;
2 1, 1;
3 1, 2;
4 1, 1, 2;
5 1, 1, 2, 1;
6 1, 2, 1, 1, 2;
7 1, 1, 2, 1, 2, 2, 1;
8 1, 2, 1, 1, 2, 1, 1, 2, 2, 1;
9 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2;
10 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2;
...
Here is the run transform iterated on the 6th row (1, 2, 1, 1, 2), which takes 6 transformations to reach 1:
1, 1, 2, 1
2, 1, 1
1, 2
1, 1
2
1
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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