A323258 -id:A323258 - cp's OEIS frontend

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

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

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 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.
We can also build the curve by successively applying the following substitution to an initial vector (1, 0):
                        .--->.
                        ^    |
                        |    v
                   .--->.    .    .--->.
                             |    ^
                             v    |
                             .--->.

Rémy Sigrist, Table of n, a(n) for n = 0..4096
Robert Ferréol (MathCurve), Saucisse de Minkowski [in French]
Rémy Sigrist, Representation of the first 1+8^4 points of the Minkowski sausage
Wikipedia, Minkowski sausage
Index entries for sequences related to coordinates of 2D curves

See A163528, A323258 and A332204 for similar sequences.

Cf. A332247 (Y-coordinates).

{ dd = [0,1,0,-1,-1,0,1,0]; z=0; for (n=0, 75, print1 (real(z)", "); z += I^vecsum(apply(d -> dd[1+d], digits(n, #dd)))) }

a(8^k-m) + a(m) = 4^k for any k >= 0 and m = 0..8^k.

A323259 a(n) is the Y-coordinate of the n-th point of the first type of Wunderlich curve (starting at the origin and occupying the first quadrant).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 4, 5, 5, 4, 3, 3, 4, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 7, 6, 6, 7, 8, 8, 7, 6, 5, 5, 5, 4, 4, 4, 3, 3, 3, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 4, 5, 5, 4, 3, 3, 4, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 10, 11, 11, 10

Offset: 1

Rémy Sigrist, Jan 09 2019

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..6561
Index entries for sequences related to coordinates of 2D curves

See A323258 for the X-coordinate and additional comments.

s = [0, 1, 2, 2+I, 1+I, I, 2*I, 1+2*I, 2+2*I];
w = apply(z -> imag(z) + I*real(z), s);
r = [0, 1, 0, 3, 2, 3, 0, 1, 0]
a(n) = {
    my (d=if (n>1, Vecrev(digits(n-1, 9)), [0]), z=s[1+d[1]]);
    for (i=2, #d,
        my (c=(3^(i-1)-1)/2*(1+I));
        z = 3^(i-1) * w[1+d[i]] + c + (z-c) * I^r[1+d[i]];
    );
    return (imag(z));
}

A323335 Square array T(n, k) read by antidiagonals upwards, n >= 0 and k >= 0: the point with coordinates X=k and Y=n is the T(n, k)-th term of the first type of Wunderlich curve.

Original entry on oeis.org

1, 2, 6, 3, 5, 7, 48, 4, 8, 16, 49, 47, 9, 15, 17, 54, 50, 46, 10, 14, 18, 55, 53, 51, 45, 11, 13, 19, 56, 60, 52, 44, 40, 12, 20, 24, 57, 59, 61, 43, 41, 39, 21, 23, 25, 462, 58, 62, 70, 42, 38, 30, 22, 26, 106, 463, 461, 63, 69, 71, 37, 31, 29, 27, 105, 107

Offset: 0

Rémy Sigrist, Jan 11 2019

Each natural numbers appears once in the sequence.

			Array T(n, k) begins:
  n\k|   0   1   2   3   4   5   6   7   8
  ---+------------------------------------
  0  |   1   6---7  16--17--18--19  24--25
     |   |   |   |   |           |   |   |
  1  |   2   5   8  15--14--13  20  23  26
     |   |   |   |           |   |   |   |
  2  |   3---4   9--10--11--12  21--22  27
     |                                   |
  3  |  48--47--46--45  40--39  30--29--28
     |   |           |   |   |   |
  4  |  49--50--51  44  41  38  31--32--33
     |           |   |   |   |           |
  5  |  54--53--52  43--42  37--36--35--34
     |   |
  6  |  55  60--61  70--71--72--73  78--79
     |   |   |   |   |           |   |   |
  7  |  56  59  62  69--68--67  74  77  80
     |   |   |   |           |   |   |   |
  8  |  57--58  63--64--65--66  75--76  81

Robert Dickau, Wunderlich Curves
Wolfram Demonstrations Project, Wunderlich Curves
Index entries for sequences that are permutations of the natural numbers

See A163334 for a similar sequence.

Cf. A323258, A323259.

T(A323259(n), A323258(n)) = n.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A323259 a(n) is the Y-coordinate of the n-th point of the first type of Wunderlich curve (starting at the origin and occupying the first quadrant).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A323335 Square array T(n, k) read by antidiagonals upwards, n >= 0 and k >= 0: the point with coordinates X=k and Y=n is the T(n, k)-th term of the first type of Wunderlich curve.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula