A341019 - OEIS

%I #14 Feb 05 2021 00:48:41

%S 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 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 11,10,11,10,9,8,9,8,9,10,11,12,11,12,13,12

%N 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 We define the family {M_n, n >= 0}, as follows:

%C - M_0 corresponds to the points (0, 0), (1, 1) and (2, 0), in that order:

%C           +

%C          / \

%C         /   \

%C        +     +

%C       O

%C - for any n >= 0, M_{n+1} is obtained by arranging 4 copies of M_n as follows:

%C                            + . . . + . . . +

%C                            .   B   .   B   .

%C        + . . . +           .       .       .

%C        .   B   .           .A     C.A     C.

%C        .       .    -->    + . . . + . . . +

%C        .A     C.           .C      .      A.

%C        + . . . +           .      B.B      .

%C       O                    .A      .      C.

%C                            + . . . + . . . +

%C                           O

%C - for any n >= 0, M_n has A087289(n) points,

%C - the space filling curve M is the limit of M_{2*n} as n tends to infinity.

%C The odd bisection of M is similar to a Hilbert's Hamiltonian walk (hence the connection with A059252).

%H Rémy Sigrist, <a href="/A341019/b341019.txt">Table of n, a(n) for n = 0..8192</a>

%H 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 Larry Riddle, <a href="http://larryriddle.agnesscott.org/ifs/spacefilling/spacefilling.htm">Space Filling Curve</a>

%H Rémy Sigrist, <a href="/A341019/a341019.gp.txt">PARI program for A341019</a>

%H <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>

%F A059252(n) = (a(2*n+1)-1)/2.

%F a(4*n) = 2*A341018(n).

%F a(16*n) = 4*a(n).

%e The curve M starts as follows:

%e        11+ 13+   +19 +21

%e         / \ / \ / \ / \

%e      10+ 12+ 14+18 +20 +22

%e         \     / \     /

%e         9+ 15+   +17 +23

%e         /     \ /     \

%e       8+  6+   +   +26 +24

%e         \ / \ 16  / \ /

%e         7+  5+   +27 +25

%e             /     \

%e           4+       +28

%e             \     /

%e         1+  3+   +29 +31

%e         / \ /     \ / \

%e       0+  2+       +30 +32

%e - so a(0) = a(2) = a(30) = a(32) = 0,

%e      a(1) = a(3) = a(29) = a(31) = 1.

%o (PARI) See Links section.

%Y Cf. A059252, A087289, A341018.

%K nonn

%O 0,5

%A _Rémy Sigrist_, Feb 02 2021