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%I #25 Sep 19 2020 02:53:46
%S 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 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 6,7,7,6,6,7,7,8,8,7,7,8,8,9,9,8,8,9,9
%N a(n) is the X-coordinate of the n-th point of the Peano curve. Sequence A332381 gives Y-coordinates.
%C This sequence is the real part of {f(n)} defined as:
%C - f(0) = 0,
%C - f(n+1) = f(n) + i^t(n)
%C where t(n) is the number of 1's and 7's minus the number of 3's and 5's
%C plus twice the number of 4's in the base 9 representation of n
%C and i denotes the imaginary unit.
%C We can also build the curve by successively applying the following substitution to an initial vector (1, 0):
%C .--->.
%C ^ |
%C | v
%C .--->/<---/--->.
%C | ^
%C v |
%C .--->.
%D 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 Rémy Sigrist, <a href="/A332380/b332380.txt">Table of n, a(n) for n = 0..6561</a>
%H 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 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 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 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%F a(9^k) = 3^k for any k >= 0.
%o (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 See A332246 for a similar sequence.
%Y Cf. A332381 (Y-coordinates).
%K nonn,look,base
%O 0,4
%A _Rémy Sigrist_, Feb 10 2020
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