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A331034 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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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?
LINKS
EXAMPLE
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 ...
+---+-------+---+---+---+---+---+---+---
...
PROG
(C++) See Links section.
CROSSREFS
See A331104 for the sequence of tile widths as they are placed.
Cf. A000002.
Sequence in context: A112505 A104638 A270648 * A057155 A037812 A037200
KEYWORD
nonn,tabl
AUTHOR
Rémy Sigrist, Jan 08 2020
STATUS
approved

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Last modified March 19 02:51 EDT 2024. Contains 370952 sequences. (Running on oeis4.)