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A335358 a(n) is the X-coordinate of the n-th point of the Koch curve; sequence A335359 gives Y-coordinates.

%I A335358 #27 Nov 12 2021 08:57:32
%S A335358 0,1,1,2,3,3,2,3,3,4,5,5,6,7,7,8,9,9,8,9,9,8,7,7,6,7,7,8,9,9,8,9,9,10,
%T A335358 11,11,12,13,13,14,15,15,16,15,15,16,17,17,18,19,19,20,21,21,20,21,21,
%U A335358 22,23,23,24,25,25,26,27,27,26,27,27,26,25,25,24
%N A335358 a(n) is the X-coordinate of the n-th point of the Koch curve; sequence A335359 gives Y-coordinates.
%C A335358 Coordinates are given on a hexagonal lattice with X-axis and Y-axis as follows:
%C A335358            Y
%C A335358           /
%C A335358          /
%C A335358         0 ---- X
%C A335358 We can build the Koch curve as follows:
%C A335358 - start from the origin,
%C A335358 - for k = 0, 1, ...:
%C A335358       - compute t as the number of 1's minus the number of 2's
%C A335358         in the base-4 representation of n (t = A065359(n)),
%C A335358       - move one step forward with an angle of t * 60 degrees
%C A335358         with respect to the X-axis.
%H A335358 Rémy Sigrist, <a href="/A335358/b335358.txt">Table of n, a(n) for n = 0..8192</a>
%H A335358 Artem Litvinov, <a href="https://docs.google.com/document/d/1FUUuR99WPY3tXgL1XXdOGkiGcfvC2eyh74WIwfZGyvA/view">A study of the Koch polyline</a>, 2021 [in Russian].
%H A335358 Wikipedia, <a href="https://en.wikipedia.org/wiki/Koch_snowflake">Koch snowflake</a>
%H A335358 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%F A335358 From _Andrey Zabolotskiy_, Nov 12 2021: (Start)
%F A335358 a(2*n) = a(n) + 2*y(n),
%F A335358 y(2*n) = a(n) - y(n),
%F A335358 where y(k) = A335359(k). See Litvinov, 2021. (End)
%e A335358 The Koch curve starts (on a hexagonal lattice) as follows:
%e A335358    .   .   .   .   .   .   +   .   .   .   .   .   .
%e A335358                           /8\
%e A335358      .   .   .   .   +---+   +---+   .   .   .   .
%e A335358                      6\  7   9  /10
%e A335358    .   .   .   +   .   +   .   +   .   +   .   .   .
%e A335358               /2\     /5        \     / \
%e A335358      .   +---+   +---+   .   .   +---+   +---+   .
%e A335358          0   1   3   4           12  13  15  16
%e A335358 Hence, a(4) = a(5) = a(7) = a(8) = 3.
%o A335358 (PARI) { hex = [1,I,I-1,-1,-I,1-I]; z=0; for (n=0, 72, print1 (real(z)", "); q=digits(n, 4); d=sum(k=1, #q, if (q[k]==1, +1, q[k]==2, -1, 0)); z+=hex[1+d%#hex]) }
%Y A335358 Cf. A065359, A335359.
%K A335358 nonn,look,base
%O A335358 0,4
%A A335358 _Rémy Sigrist_, Jun 03 2020