This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A330844 #53 Aug 23 2024 16:04:34
%S A330844 1,6,18,42,90,186,366,690,1278,2322,4182,7482,13326,23682,42006,74442,
%T A330844 131838,233394,413094,731034,1293582,2288898,4049910,7165674,12678366,
%U A330844 22431954,39688902,70221498,124242606,219821730,388929174,688129674,1217502846,2154118770
%N 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.
%C A330844 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).
%C A330844 A Koch snowflake tile at step n+1 touches one or two Koch snowflakes tile at step n.
%C A330844 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).
%C A330844 We can also compute the sequence by mean of necklaces:
%C A330844 - let h_2 = 0 and K_2 = (111111) (this is a necklace with six 1's),
%C A330844 - for n = 2, 3, ...:
%C A330844 h_{n+1} = h_n + the number of occurrences of "100001" in K_n
%C A330844 apply the following substitutions to K_n, in that order:
%C A330844 - "100001" --> "00" (this loop will come off and form a hole),
%C A330844 - "1" --> "110",
%C A330844 - K_{n+1} is the necklace obtained after these substitutions,
%C A330844 - now we have a(n) = #K_n + 6 * h_n,
%C A330844 - the sequence {h_n} corresponds to the number of "holes" in the construction,
%C A330844 - the sequence {K_n} encodes the configuration of the tiles at n-th step (excluding holes),
%C A330844 - we can recover this configuration, up to some rotation, as follows:
%C A330844 - start from the origin pointing to the right:
%C A330844 - for m = 1..#K_n:
%C A330844 - place a Koch snowflake with an arm pointing to the right,
%C A330844 - move one step forward,
%C A330844 - if K_n(m) = 1 then turn 60 degrees to the left,
%C A330844 - otherwise turn 60 degrees to the right,
%C A330844 - at the end, we are at the origin again.
%C A330844 Apparently, the number of holes is related to A077879 in the following manner:
%C A330844 - h_{n+1} - h_n = 6*A077879(n-6) for n >= 6.
%H A330844 Aperiodic Rhomb Tiling, <a href="http://www.hibma.org/wpaperiodictiling/index.php/2x2-supertiles/koch-tiles/">Koch Tiles</a>
%H A330844 Ben Van Dusen, Billy Scannel, Richard Taylor, <a href="https://arxiv.org/abs/1209.2180">A Fractal Comparison of Escher and Koch Tessellations</a>, arXiv:1209.2180 [physics.pop-ph], 2012-2013.
%H A330844 Rémy Sigrist, <a href="/A330844/a330844.png">Two adjacent Koch snowflakes at consecutive steps</a>
%H A330844 Rémy Sigrist, <a href="/A330844/a330844_1.png">Representation of the bounding hexagons for the first 10 steps</a>
%H A330844 Rémy Sigrist, <a href="/A330844/a330844_2.png">Representation of the figure after 13 steps</a>
%H A330844 Rémy Sigrist, <a href="/A330844/a330844.txt">C# program for A330844</a>
%H A330844 Rémy Sigrist, <a href="/A330844/a330844_1.txt">C++ program for A330844</a>
%H A330844 Spacefilling Curves, <a href="https://spacefillingcurves.wordpress.com/2019/09/01/on-filling-the-koch-snowflake/">On filling the Koch Snowflake</a>
%H A330844 Wikipedia, <a href="https://en.wikipedia.org/wiki/Koch_snowflake">Koch snowflake</a>
%H A330844 Wikipedia, <a href="https://en.wikipedia.org/wiki/Trihexagonal_tiling">Trihexagonal tiling</a>
%o A330844 (C#) // See Links section.
%o A330844 (C++) // See Links section.
%Y A330844 Cf. A077879, A321257.
%K A330844 nonn
%O A330844 1,2
%A A330844 _Rémy Sigrist_, Feb 16 2020