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A362228 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.

%I A362228 #53 Apr 27 2025 03:22:46
%S A362228 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,
%T A362228 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,
%U A362228 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
%N A362228 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.
%C A362228 The run transform replaces each run of consecutive identical values with a single value representing the length of that run.
%C A362228 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.
%C A362228 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
%e A362228 The triangle begins:
%e A362228    0   1;
%e A362228    1   2;
%e A362228    2   1, 1;
%e A362228    3   1, 2;
%e A362228    4   1, 1, 2;
%e A362228    5   1, 1, 2, 1;
%e A362228    6   1, 2, 1, 1, 2;
%e A362228    7   1, 1, 2, 1, 2, 2, 1;
%e A362228    8   1, 2, 1, 1, 2, 1, 1, 2, 2, 1;
%e A362228    9   1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2;
%e A362228   10   1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2;
%e A362228        ...
%e A362228 Here is the run transform iterated on the 6th row (1, 2, 1, 1, 2), which takes 6 transformations to reach 1:
%e A362228    1, 1, 2, 1
%e A362228    2, 1, 1
%e A362228    1, 2
%e A362228    1, 1
%e A362228    2
%e A362228    1
%Y A362228 Cf. A327662.
%K A362228 nonn,tabf
%O A362228 0,2
%A A362228 _Neal Gersh Tolunsky_, Apr 11 2023