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.

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

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

Table of n, a(n) for n=1..87.

Rémy Sigrist, Scatterplot of twice the number of 1's minus the number of 2's among the first n terms for n = 1..200010001

Rémy Sigrist, C++ program for A331034

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

Adjacent sequences: A331031 A331032 A331033 * A331035 A331036 A331037

Keyword

nonn,tabl,changed

Author

Rémy Sigrist, Jan 08 2020

Status

approved