A332246 -id:A332246 - cp's OEIS frontend

A332247 a(n) is the Y-coordinate of the n-th point of the Minkowski sausage (or Minkowski curve). Sequence A332246 gives X-coordinates.

0, 0, 1, 1, 0, -1, -1, 0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 4, 3, 3, 4, 4, 3, 3, 2, 2, 2, 1, 1, 0, -1, -1, -2, -2, -2, -3, -3, -4, -4, -3, -3, -4, -5, -5, -4, -4, -3, -3, -2, -2, -2, -1, -1, 0, 0, 1, 1, 0, -1, -1, 0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 3, 3, 4, 5

Offset: 0

Rémy Sigrist, Feb 08 2020

This sequence is the imaginary 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 6's minus the number of 3's and 4's
             in the base 8 representation of n
    and i denotes the imaginary unit.

Rémy Sigrist, Table of n, a(n) for n = 0..4096
Robert Ferréol (MathCurve), Saucisse de Minkowski [in French]
Wikipedia, Minkowski sausage
Index entries for sequences related to coordinates of 2D curves

Cf. A332246 (X-coordinates and additional comments).

{ dd = [0,1,0,-1,-1,0,1,0]; z=0; for (n=0, 77, print1 (imag(z)", "); z += I^vecsum(apply(d -> dd[1+d], digits(n, #dd)))) }

a(8^k-m) = -a(m) for any k >= 0 and m = 0..8^k.

A332249 a(n) is the X-coordinate of the n-th point of the quadratic Koch curve. Sequence A332250 gives Y-coordinates.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 6, 6, 7, 7, 8, 8, 9, 9, 8, 8, 9, 9, 8, 8, 7, 7, 6, 6, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 15, 15, 16, 16, 17, 17, 18, 18, 19

Offset: 0

Rémy Sigrist, Feb 08 2020

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 minus the number of 3's
             in the base 5 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
                   .--->.    .--->.

Rémy Sigrist, Table of n, a(n) for n = 0..15625
Robert Ferréol (MathCurve), Courbe de Koch quadratique [in French]
Rémy Sigrist, Representation of the first 1+2^10 points of the quadratic Koch curve
Index entries for sequences related to coordinates of 2D curves

See A332246 for a similar sequence.

Cf. A229217, A332250 (Y-coordinates).

{ k = [0, 1, 0, -1, 0]; z=0; for (n=0, 77, print1 (real(z) ", "); z += I^vecsum(apply(d -> k[1+d], digits(n, #k)))) }

a(5^k-m) + a(m) = 3^k for any k >= 0 and m = 0..5^k.

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

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Feb 10 2020

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
       .--->/<---/--->.
            |    ^
            v    |
            .--->.

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.

Rémy Sigrist, Table of n, a(n) for n = 0..6561
Joerg Arndt, Plane-filling curves on all uniform grids, arXiv:1607.02433 [math.CO], 2016, 2018. Curve R9-1 drawn in figure 4.1-O (top row forms, vertical mirror image).
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.
Index entries for sequences related to coordinates of 2D curves

See A332246 for a similar sequence.

Cf. A332381 (Y-coordinates).

{ [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)))) }

a(9^k) = 3^k for any k >= 0.

cp's OEIS Frontend

A332247 a(n) is the Y-coordinate of the n-th point of the Minkowski sausage (or Minkowski curve). Sequence A332246 gives X-coordinates.

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A332249 a(n) is the X-coordinate of the n-th point of the quadratic Koch curve. Sequence A332250 gives Y-coordinates.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

PARI

Formula