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A332380
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a(n) is the X-coordinate of the n-th point of the Peano curve. Sequence A332381 gives Y-coordinates.
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2
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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, 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, 6, 7, 7, 6, 6, 7, 7, 8, 8, 7, 7, 8, 8, 9, 9, 8, 8, 9, 9
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OFFSET
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0,4
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COMMENTS
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This sequence is the real part of {f(n)} defined as:
- f(0) = 0,
- f(n+1) = f(n) + i^t(n)
where t(n) is the number of 1's and 7's minus the number of 3's and 5's
plus twice the number of 4's in the base 9 representation of n
and i denotes the imaginary unit.
We can also build the curve by successively applying the following substitution to an initial vector (1, 0):
.--->.
^ |
| v
.--->/<---/--->.
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.--->.
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REFERENCES
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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.
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LINKS
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Donald E. Knuth, Selected Papers on Fun and Games, 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.
Walter Wunderlich, Über Peano-Kurven, 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.
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FORMULA
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a(9^k) = 3^k for any k >= 0.
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PROG
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(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)))) }
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CROSSREFS
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See A332246 for a similar sequence.
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KEYWORD
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AUTHOR
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STATUS
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approved
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