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A331034
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Consider an empty square array visited by antidiagonals downwards; when the current square is empty, place a 1 X 1 tile or a 1 X 2 tile on it (without overlapping any prior tile) such that the RUNS transform of the sequence of tile widths as they are placed on the grid equals the sequence of tile widths as they are visited; first place a 1 X 1 tile. This sequence corresponds to the tile widths as they are visited.
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3
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1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1
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OFFSET
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1,2
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COMMENTS
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This sequence is a 2-dimensional variant of Kolakoski sequence (A000002).
The sequence {a(n}} gives the width of the tiles as they are visited, while, say {b(n)}, gives the width of the tiles as they are placed; so:
- a 1 X 2 tile will account for two terms in {a(n)} and for one term in {b(n)},
- a 1 X 1 tile will account for one term in both sequences,
- {a(n)} = RUNS({b(n)}).
Is the density of 1's equal to 1/3?
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LINKS
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EXAMPLE
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Square array begins:
+---+-------+---+---+-------+---+---+---
| 1 | 2 2 | 1 | 1 | 2 2 | 1 | 1 | 2 ...
+---+---+---+---+---+---+---+---+---+---
| 2 2 | 2 2 | 2 2 | 2 2 | 1 | 1 ...
+---+---+---+---+---+---+---+---+---+---
| 1 | 2 2 | 1 | 1 | 2 2 | 1 | 2 2 ...
+---+-------+---+---+---+---+---+---+---
| 1 | 2 2 | 2 2 | 1 | 2 2 | 1 | 1 ...
+---+---+---+---+---+---+-------+---+---
| 1 | 1 | 2 2 | 2 2 | 2 2 | 2 2 ...
+---+---+---+---+-------+---+---+---+---
| 2 2 | 1 | 1 | 2 2 | 1 | 2 2 | 1 ...
+---+---+---+---+---+---+---+-------+---
| 1 | 2 2 | 1 | 1 | 2 2 | 2 2 | 1 ...
+---+---+---+---+---+---+---+---+---+---
| 2 2 | 1 | 1 | 2 2 | 1 | 1 | 2 2 ...
+---+---+---+---+---+---+---+---+-------
| 1 | 2 2 | 1 | 1 | 1 | 2 2 | 2 2 ...
+---+-------+---+---+---+---+---+---+---
...
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PROG
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(C++) See Links section.
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CROSSREFS
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See A331104 for the sequence of tile widths as they are placed.
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KEYWORD
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AUTHOR
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STATUS
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approved
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