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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.
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%I #20 Feb 05 2021 00:48:33

%S 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 8,9,10,11,12,11,12,13,14,15,14,15,14,13,12,13,14,15,14,15,14,13,12,

%U 11,12,11,10,9,8,9,8,9,8,9,10,11,12,11,12,13,14,15

%N 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 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 A059253, see illustration in Links section).

%H Rémy Sigrist, <a href="/A341018/b341018.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="/A341018/a341018.png">Illustration of M_6</a>

%H Rémy Sigrist, <a href="/A341018/a341018_1.png">Illustration of the connection between the space filling curve M and Hilbert Hamiltonian walk</a>

%H Rémy Sigrist, <a href="/A341018/a341018.gp.txt">PARI program for A341018</a>

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

%F a(n) = A341019(n) iff n belongs to A000695.

%F a(2*n-1) + A341019(2*n-1) = a(2*n) + A341019(2*n) for any n > 0.

%F a(2*n) - A341019(2*n) = a(2*n+1) - A341019(2*n+1) for any n >= 0.

%F A059253(n) = (a(2*n+1) - 1)/2.

%F a(4*n) = 2*A341019(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(8) = a(10) = 0,

%e a(1) = a(7) = a(9) = a(11) = 1.

%o (PARI) See Links section.

%Y Cf. A000695, A059253, A087289, A341019.

%K nonn

%O 0,3

%A _Rémy Sigrist_, Feb 02 2021