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%I #17 Feb 09 2021 22:01:43
%S 0,1,3,2,0,1,3,4,6,5,3,4,6,5,3,2,0,1,3,2,0,1,3,4,6,7,9,8,6,7,9,10,12,
%T 11,9,10,12,11,9,8,6,7,9,8,6,7,9,10,12,11,9,10,12,11,9,8,6,5,3,4,6,5,
%U 3,2,0,1,3,2,0,1,3,4,6,5,3,4,6,5,3,2,0,1,3
%N a(n) is the X-coordinate of the n-th point of the space filling curve A defined in Comments section; A341164 gives Y-coordinates.
%C Coordinates are given on a hexagonal lattice with X-axis and Y-axis as follows:
%C Y
%C /
%C /
%C 0 ---- X
%C We define the family {A_n, n >= 0} as follows:
%C - A_0 corresponds to the points (0, 0), (1, 1) and (3, 0), in that order:
%C . __+__ .
%C __---- ----__
%C + . . +
%C 0
%C - for any n >= 0, A_{n+1} is obtained by arranging 4 copies of A_n as follows:
%C +
%C /B\
%C + / \
%C /B\ /A C\
%C / \ --> +-------+
%C /A C\ /B\C B/A\
%C +-------+ / \ / \
%C O /A C\A/B C\
%C +-------+-------+
%C O
%C - the space filling curve A is the limit of A_n as n tends to infinity.
%C This sequence has similarities with A341018.
%H Rémy Sigrist, <a href="/A341163/b341163.txt">Table of n, a(n) for n = 0..8192</a>
%H Zbigniew Fiedorowicz, <a href="https://people.math.osu.edu/fiedorowicz.1/math655/peano_t.html">The Peano Curve Theorem</a>
%H Rémy Sigrist, <a href="/A341163/a341163.png">Illustration of A_5</a>
%H Rémy Sigrist, <a href="/A341163/a341163_1.png">Illustration of the first bisection of A</a>
%H Rémy Sigrist, <a href="/A341163/a341163_2.png">Illustration of the first quadrisection of A</a>
%H Rémy Sigrist, <a href="/A341163/a341163_3.png">Illustration of the fourth quadrisection of A</a>
%H Rémy Sigrist, <a href="/A341163/a341163.gp.txt">PARI program for A341163</a>
%H <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%e The curve A starts as follows:
%e .
%e . .
%e . 5 .
%e 4 . . 6
%e . . 3 . .
%e . 1 . . 7 .
%e 0 . . 2 . . 8
%e - so a(0) = a(4) = 0,
%e a(1) = a(5) = 1,
%e a(3) = 2,
%e a(2) = a(6) = 3,
%e a(8) = 6.
%o (PARI) See Links section.
%Y Cf. A341018, A341164.
%K nonn,look
%O 0,3
%A _Rémy Sigrist_, Feb 06 2021
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