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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.
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