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

Author

Rémy Sigrist, Feb 08 2020

Keywords

• nonn
• base

Comments

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 |

.--->.

Links

• 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

Crossrefs

See A163528, A323258 and A332204 for similar sequences.

Cf. A332247 (Y-coordinates).

Programs

PARI

{ 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)))) }

Formula

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