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%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
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