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A342425 a(n) is the number of (connected) tiles at distance n from the leftmost tile in the Hofstetter-4fold tiling. 3
1, 3, 6, 6, 8, 13, 12, 14, 15, 17, 22, 21, 23, 25, 28, 29, 29, 32, 36, 37, 36, 42, 42, 40, 45, 47, 51, 51, 51, 55, 57, 57, 58, 58, 69, 66, 63, 73, 73, 72, 72, 72, 84, 75, 84, 88, 85, 87, 86, 91, 98, 94, 96, 100, 107, 103, 100, 105, 113, 110, 110, 115, 115, 114 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

We build the Hofstetter-4fold tiling as follows:

- H_0 corresponds to a 2 X 2 square:

      +---+---+

      |       |

      +       +

      |       |

      +---+---+

     O

- for any k >= 0, H_{k+1} is obtained by arranging 4 copies of H_k, rotated by 0, 90, 180, 270 degrees clockwise respectively, as follows:

                            +.......+

                            .     90.

      +.......+         +.......+   ....+

      .       .         .0      .   .   .

      .       .   -->   .       .....   .

      .       .         .       .    180.

      +.......+         +.......+.......+

     O                 O    .270    .

                            +.......+

- note that:

   - the copy rotated by 0 degrees hides some squares on the copies rotated by 90 and 270 degrees,

   - the copy rotated by 90 degrees hides some squares on the copy rotated by 180 degrees,

   - the copy rotated by 180 degrees hides some squares on the copy rotated by 270 degrees,

- the Hofstetter-4fold tiling corresponds to the limit of H_k as k tends to infinity,

- in this sequence we consider only connected tiles (whose squares are vertically or horizontally adjacent); disconnected tiles made up of two diagonally adjacent squares are counted as two distinct connected tiles.

LINKS

Rémy Sigrist, Table of n, a(n) for n = 0..5000

Rémy Sigrist, Illustration of initial terms

Rémy Sigrist, Colored representation of the tiles at distance <= 512 (where the color is function of the distance)

Rémy Sigrist, C# program for A342425

Tilings Encyclopedia, Hofstetter-4fold (plain)

EXAMPLE

See illustration in Links section.

PROG

(C#) See Links section.

CROSSREFS

Cf. A342577, A342597.

Sequence in context: A229986 A025500 A141218 * A318845 A147866 A135702

Adjacent sequences:  A342422 A342423 A342424 * A342426 A342427 A342428

KEYWORD

nonn

AUTHOR

Rémy Sigrist, Mar 11 2021

STATUS

approved

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Last modified July 25 03:53 EDT 2021. Contains 346283 sequences. (Running on oeis4.)