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A341164 a(n) is the Y-coordinate of the n-th point of the space filling curve A defined in Comments section; A341163 gives X-coordinates.

%I A341164 #14 Feb 09 2021 22:02:30
%S A341164 0,1,0,2,3,4,3,1,0,2,3,4,3,5,6,5,6,7,6,8,9,10,9,7,6,4,3,2,3,1,0,1,0,2,
%T A341164 3,4,3,5,6,5,6,7,6,8,9,10,9,7,6,8,9,10,9,11,12,11,12,11,12,10,9,8,9,
%U A341164 11,12,13,12,14,15,16,15,13,12,14,15,16,15,17
%N A341164 a(n) is the Y-coordinate of the n-th point of the space filling curve A defined in Comments section; A341163 gives X-coordinates.
%C A341164 Coordinates are given on a hexagonal lattice with X-axis and Y-axis as follows:
%C A341164            Y
%C A341164           /
%C A341164          /
%C A341164         0 ---- X
%C A341164 We define the family {A_n, n >= 0} as follows:
%C A341164 - A_0 corresponds to the points (0, 0), (1, 1) and (3, 0), in that order:
%C A341164            .   __+__   .
%C A341164          __----     ----__
%C A341164         +     .     .     +
%C A341164        0
%C A341164 - for any n >= 0, A_{n+1} is obtained by arranging 4 copies of A_n as follows:
%C A341164                          +
%C A341164                         /B\
%C A341164         +              /   \
%C A341164        /B\            /A   C\
%C A341164       /   \    -->   +-------+
%C A341164      /A   C\        /B\C   B/A\
%C A341164     +-------+      /   \   /   \
%C A341164    O              /A   C\A/B   C\
%C A341164                  +-------+-------+
%C A341164                 O
%C A341164 - the space filling curve A is the limit of A_n as n tends to infinity.
%H A341164 Rémy Sigrist, <a href="/A341164/b341164.txt">Table of n, a(n) for n = 0..8192</a>
%H A341164 Zbigniew Fiedorowicz, <a href="https://people.math.osu.edu/fiedorowicz.1/math655/peano_t.html">The Peano Curve Theorem</a>
%H A341164 Rémy Sigrist, <a href="/A341164/a341164.gp.txt">PARI program for A341164</a>
%H A341164 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%e A341164 The curve A starts as follows:
%e A341164                   .
%e A341164                 .   .
%e A341164               .   5   .
%e A341164             4   .   .   6
%e A341164           .   .   3   .   .
%e A341164         .   1   .   .   7   .
%e A341164       0   .   .   2   .   .   8
%e A341164 - so a(0) = a(2) = a(8) = 0,
%e A341164      a(1) = a(7) = 1,
%e A341164      a(3) = 2,
%e A341164      a(4) = a(6) = 3,
%e A341164      a(5) = 4.
%o A341164 (PARI) See Links section.
%Y A341164 Cf. A341163.
%K A341164 nonn,look
%O A341164 0,4
%A A341164 _Rémy Sigrist_, Feb 06 2021