A342425 a(n) is the number of (connected) tiles at distance n from the leftmost tile in the Hofstetter-4fold tiling.

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

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