A327662 -id:A327662 - cp's OEIS frontend

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.

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

Offset: 0

Neal Gersh Tolunsky, Apr 11 2023

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

			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

Cf. A327662.

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Crossrefs