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A332380 a(n) is the X-coordinate of the n-th point of the Peano curve. Sequence A332381 gives Y-coordinates.

%I A332380 #25 Sep 19 2020 02:53:46
%S A332380 0,1,1,2,2,1,1,2,2,3,3,2,2,3,3,4,4,3,3,4,4,5,5,4,4,5,5,6,6,7,7,6,6,5,
%T A332380 5,6,6,5,5,4,4,5,5,4,4,3,3,4,4,3,3,2,2,3,3,4,4,5,5,4,4,5,5,6,6,5,5,6,
%U A332380 6,7,7,6,6,7,7,8,8,7,7,8,8,9,9,8,8,9,9
%N A332380 a(n) is the X-coordinate of the n-th point of the Peano curve. Sequence A332381 gives Y-coordinates.
%C A332380 This sequence is the real part of {f(n)} defined as:
%C A332380 - f(0) = 0,
%C A332380 - f(n+1) = f(n) + i^t(n)
%C A332380   where t(n) is the number of 1's and 7's minus the number of 3's and 5's
%C A332380   plus twice the number of 4's in the base 9 representation of n
%C A332380   and i denotes the imaginary unit.
%C A332380 We can also build the curve by successively applying the following substitution to an initial vector (1, 0):
%C A332380             .--->.
%C A332380             ^    |
%C A332380             |    v
%C A332380        .--->/<---/--->.
%C A332380             |    ^
%C A332380             v    |
%C A332380             .--->.
%D A332380 Benoit B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Co., 1983, section 7, "Harnessing the Peano Monster Curves", page 62 description and plate 63 bottom right drawn with chamfered corners.
%H A332380 Rémy Sigrist, <a href="/A332380/b332380.txt">Table of n, a(n) for n = 0..6561</a>
%H A332380 Joerg Arndt, <a href="https://arxiv.org/abs/1607.02433">Plane-filling curves on all uniform grids</a>, arXiv:1607.02433 [math.CO], 2016, 2018.  Curve R9-1 drawn in figure 4.1-O (top row forms, vertical mirror image).
%H A332380 Donald E. Knuth, <a href="http://www-cs-faculty.stanford.edu/~uno/fg.html">Selected Papers on Fun and Games</a>, CSLI Lecture Notes Number 192, CSLI Publications, 2010, ISBN 978-1-57586-585-0, page 611 folding product DUUUDDDU drawn at 45 degrees in a labyrinth.
%H A332380 Walter Wunderlich, <a href="http://sodwana.uni-ak.ac.at/geom/mitarbeiter/wallner/wunderlich/pdf/125.pdf">Über Peano-Kurven</a>, Elemente der Mathematik, volume 28, number 1, 1973, pages 1-10.  See section 4 serpentine type 010 101 010 as illustrated in figure 3, the coordinates here being diagonal steps across the unit squares there.
%H A332380 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%F A332380 a(9^k) = 3^k for any k >= 0.
%o A332380 (PARI) { [R,U,L,D]=[0..3]; p = [R,U,R,D,L,D,R,U,R]; z=0; for (n=0, 86, print1 (real(z) ", "); z += I^vecsum(apply(d -> p[1+d], digits(n, #p)))) }
%Y A332380 See A332246 for a similar sequence.
%Y A332380 Cf. A332381 (Y-coordinates).
%K A332380 nonn,look,base
%O A332380 0,4
%A A332380 _Rémy Sigrist_, Feb 10 2020