A330844 Start with a Koch snowflake tile, and repeatedly append along the tiles of the previous step tiles scaled by a factor of sqrt(1/3) and rotated by 90 degrees that do not overlap with any prior tile and of which one third of the perimeter matches one sixth of the perimeter of adjacent prior tiles; a(n) gives the number of tiles appended at n-th step.

1, 6, 18, 42, 90, 186, 366, 690, 1278, 2322, 4182, 7482, 13326, 23682, 42006, 74442, 131838, 233394, 413094, 731034, 1293582, 2288898, 4049910, 7165674, 12678366, 22431954, 39688902, 70221498, 124242606, 219821730, 388929174, 688129674, 1217502846, 2154118770

Offset: 1

Author

Rémy Sigrist, Feb 16 2020

Keywords

• nonn

Comments

One continuous third of the perimeter of a Koch snowflake tile at step n+1 coincides with one sixth of the perimeter of an adjacent Koch snowflake tile at step n; this is the maximum matching that can be achieved with the given scaling and rotation parameters (see illustration of two adjacent Koch snowflakes at consecutive steps in Links section).

A Koch snowflake tile at step n+1 touches one or two Koch snowflakes tile at step n.

The bounding hexagons of the Koch snowflake tiles at step n lie on a trihexagonal tiling (see representation of the bounding hexagons in Links sections; hexagons of the same color lie on the same trihexagonal tiling).

We can also compute the sequence by mean of necklaces:

- let h_2 = 0 and K_2 = (111111) (this is a necklace with six 1's),

- for n = 2, 3, ...:

h_{n+1} = h_n + the number of occurrences of "100001" in K_n

apply the following substitutions to K_n, in that order:

- "100001" --> "00" (this loop will come off and form a hole),

- "1" --> "110",

- K_{n+1} is the necklace obtained after these substitutions,

- now we have a(n) = #K_n + 6 * h_n,

- the sequence {h_n} corresponds to the number of "holes" in the construction,

- the sequence {K_n} encodes the configuration of the tiles at n-th step (excluding holes),

- we can recover this configuration, up to some rotation, as follows:

- start from the origin pointing to the right:

- for m = 1..#K_n:

- place a Koch snowflake with an arm pointing to the right,

- move one step forward,

- if K_n(m) = 1 then turn 60 degrees to the left,

- otherwise turn 60 degrees to the right,

- at the end, we are at the origin again.

Apparently, the number of holes is related to A077879 in the following manner:

- h_{n+1} - h_n = 6*A077879(n-6) for n >= 6.

Links

• Aperiodic Rhomb Tiling, Koch Tiles
• Ben Van Dusen, Billy Scannel, Richard Taylor, A Fractal Comparison of Escher and Koch Tessellations, arXiv:1209.2180 [physics.pop-ph], 2012-2013.
• Rémy Sigrist, Two adjacent Koch snowflakes at consecutive steps
• Rémy Sigrist, Representation of the bounding hexagons for the first 10 steps
• Rémy Sigrist, Representation of the figure after 13 steps
• Rémy Sigrist, C# program for A330844
• Rémy Sigrist, C++ program for A330844
• Spacefilling Curves, On filling the Koch Snowflake
• Wikipedia, Koch snowflake
• Wikipedia, Trihexagonal tiling

Crossrefs

Cf. A077879, A321257.