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

%I A341019 #14 Feb 05 2021 00:48:41
%S A341019 0,1,0,1,2,3,4,3,4,5,6,7,6,7,6,5,4,5,6,7,6,7,6,5,4,3,4,3,2,1,0,1,0,1,
%T A341019 2,3,2,3,2,1,0,1,0,1,2,3,4,3,4,5,4,5,6,7,8,7,8,7,6,5,6,5,6,7,8,9,10,
%U A341019 11,10,11,10,9,8,9,8,9,10,11,12,11,12,13,12
%N A341019 a(n) is the Y-coordinate of the n-th point of the space filling curve M defined in Comments section; A341018 gives X-coordinates.
%C A341019 We define the family {M_n, n >= 0}, as follows:
%C A341019 - M_0 corresponds to the points (0, 0), (1, 1) and (2, 0), in that order:
%C A341019           +
%C A341019          / \
%C A341019         /   \
%C A341019        +     +
%C A341019       O
%C A341019 - for any n >= 0, M_{n+1} is obtained by arranging 4 copies of M_n as follows:
%C A341019                            + . . . + . . . +
%C A341019                            .   B   .   B   .
%C A341019        + . . . +           .       .       .
%C A341019        .   B   .           .A     C.A     C.
%C A341019        .       .    -->    + . . . + . . . +
%C A341019        .A     C.           .C      .      A.
%C A341019        + . . . +           .      B.B      .
%C A341019       O                    .A      .      C.
%C A341019                            + . . . + . . . +
%C A341019                           O
%C A341019 - for any n >= 0, M_n has A087289(n) points,
%C A341019 - the space filling curve M is the limit of M_{2*n} as n tends to infinity.
%C A341019 The odd bisection of M is similar to a Hilbert's Hamiltonian walk (hence the connection with A059252).
%H A341019 Rémy Sigrist, <a href="/A341019/b341019.txt">Table of n, a(n) for n = 0..8192</a>
%H A341019 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 A341019 Larry Riddle, <a href="http://larryriddle.agnesscott.org/ifs/spacefilling/spacefilling.htm">Space Filling Curve</a>
%H A341019 Rémy Sigrist, <a href="/A341019/a341019.gp.txt">PARI program for A341019</a>
%H A341019 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%F A341019 A059252(n) = (a(2*n+1)-1)/2.
%F A341019 a(4*n) = 2*A341018(n).
%F A341019 a(16*n) = 4*a(n).
%e A341019 The curve M starts as follows:
%e A341019        11+ 13+   +19 +21
%e A341019         / \ / \ / \ / \
%e A341019      10+ 12+ 14+18 +20 +22
%e A341019         \     / \     /
%e A341019         9+ 15+   +17 +23
%e A341019         /     \ /     \
%e A341019       8+  6+   +   +26 +24
%e A341019         \ / \ 16  / \ /
%e A341019         7+  5+   +27 +25
%e A341019             /     \
%e A341019           4+       +28
%e A341019             \     /
%e A341019         1+  3+   +29 +31
%e A341019         / \ /     \ / \
%e A341019       0+  2+       +30 +32
%e A341019 - so a(0) = a(2) = a(30) = a(32) = 0,
%e A341019      a(1) = a(3) = a(29) = a(31) = 1.
%o A341019 (PARI) See Links section.
%Y A341019 Cf. A059252, A087289, A341018.
%K A341019 nonn
%O A341019 0,5
%A A341019 _Rémy Sigrist_, Feb 02 2021