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A341018 a(n) is the X-coordinate of the n-th point of the space filling curve M defined in Comments section; A341019 gives Y-coordinates.

%I A341018 #20 Feb 05 2021 00:48:33
%S A341018 0,1,2,3,2,3,2,1,0,1,0,1,2,3,4,3,4,5,4,5,6,7,8,7,8,7,6,5,6,5,6,7,8,9,
%T A341018 8,9,10,11,12,11,12,13,14,15,14,15,14,13,12,13,14,15,14,15,14,13,12,
%U A341018 11,12,11,10,9,8,9,8,9,8,9,10,11,12,11,12,13,14,15
%N A341018 a(n) is the X-coordinate of the n-th point of the space filling curve M defined in Comments section; A341019 gives Y-coordinates.
%C A341018 We define the family {M_n, n >= 0}, as follows:
%C A341018 - M_0 corresponds to the points (0, 0), (1, 1) and (2, 0), in that order:
%C A341018           +
%C A341018          / \
%C A341018         /   \
%C A341018        +     +
%C A341018       O
%C A341018 - for any n >= 0, M_{n+1} is obtained by arranging 4 copies of M_n as follows:
%C A341018                            + . . . + . . . +
%C A341018                            .   B   .   B   .
%C A341018        + . . . +           .       .       .
%C A341018        .   B   .           .A     C.A     C.
%C A341018        .       .    -->    + . . . + . . . +
%C A341018        .A     C.           .C      .      A.
%C A341018        + . . . +           .      B.B      .
%C A341018       O                    .A      .      C.
%C A341018                            + . . . + . . . +
%C A341018                           O
%C A341018 - for any n >= 0, M_n has A087289(n) points,
%C A341018 - the space filling curve M is the limit of M_{2*n} as n tends to infinity.
%C A341018 The odd bisection of M is similar to a Hilbert's Hamiltonian walk (hence the connection with A059253, see illustration in Links section).
%H A341018 Rémy Sigrist, <a href="/A341018/b341018.txt">Table of n, a(n) for n = 0..8192</a>
%H A341018 F. M. Dekking, <a href="http://dx.doi.org/10.1016/0001-8708(82)90066-4">Recurrent Sets</a>, Advances in Mathematics, vol. 44, no. 1, 1982.
%H A341018 Larry Riddle, <a href="http://larryriddle.agnesscott.org/ifs/spacefilling/spacefilling.htm">Space Filling Curve</a>
%H A341018 Rémy Sigrist, <a href="/A341018/a341018.png">Illustration of M_6</a>
%H A341018 Rémy Sigrist, <a href="/A341018/a341018_1.png">Illustration of the connection between the space filling curve M and Hilbert Hamiltonian walk</a>
%H A341018 Rémy Sigrist, <a href="/A341018/a341018.gp.txt">PARI program for A341018</a>
%H A341018 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%F A341018 a(n) = A341019(n) iff n belongs to A000695.
%F A341018 a(2*n-1) + A341019(2*n-1) = a(2*n) + A341019(2*n) for any n > 0.
%F A341018 a(2*n) - A341019(2*n) = a(2*n+1) - A341019(2*n+1) for any n >= 0.
%F A341018 A059253(n) = (a(2*n+1) - 1)/2.
%F A341018 a(4*n) = 2*A341019(n).
%F A341018 a(16*n) = 4*a(n).
%e A341018 The curve M starts as follows:
%e A341018        11+ 13+   +19 +21
%e A341018         / \ / \ / \ / \
%e A341018      10+ 12+ 14+18 +20 +22
%e A341018         \     / \     /
%e A341018         9+ 15+   +17 +23
%e A341018         /     \ /     \
%e A341018       8+  6+   +   +26 +24
%e A341018         \ / \ 16  / \ /
%e A341018         7+  5+   +27 +25
%e A341018             /     \
%e A341018           4+       +28
%e A341018             \     /
%e A341018         1+  3+   +29 +31
%e A341018         / \ /     \ / \
%e A341018       0+  2+       +30 +32
%e A341018 - so a(0) = a(8) = a(10) = 0,
%e A341018      a(1) = a(7) = a(9) = a(11) = 1.
%o A341018 (PARI) See Links section.
%Y A341018 Cf. A000695, A059253, A087289, A341019.
%K A341018 nonn
%O A341018 0,3
%A A341018 _Rémy Sigrist_, Feb 02 2021