A332204 -id:A332204 - 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.

A332205 a(n) is the imaginary part of f(n) defined by f(0) = 0, and f(n+1) = f(n) + g((1+i)^(A065359(n) mod 8)) (where g(z) = z/gcd(Re(z), Im(z)) and i denotes the imaginary unit).

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Feb 07 2020

Looks much like A005536, in particular in respect of its symmetries of scale (compare the scatterplots). - Peter Munn, Jun 21 2021

Rémy Sigrist, Table of n, a(n) for n = 0..16384
Larry Riddle, Koch Curve
Rémy Sigrist, PARI program for A332205
Index entries for sequences related to coordinates of 2D curves

Cf. A005536, A007052, A065359, A332204 (real part and additional comments), A332206 (positions of 0's, cf. A001196).

A065359[0] = 0;
A065359[n_] := -Total[(-1)^PositionIndex[Reverse[IntegerDigits[n, 2]]][1]];
g[z_] := z/GCD[Re[z], Im[z]];
Module[{n = 0}, Im[NestList[# + g[(1+I)^A065359[n++]] &, 0, 100]]] (* Paolo Xausa, Aug 28 2024 *)

PARI
```
\\ See Links section.
```

a(2^(2*k-1)) = A007052(k) for any k >= 0.

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

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