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a(n) is the X-coordinate of the n-th point of the Minkowski sausage (or Minkowski curve). Sequence A332247 gives Y-coordinates.
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%I #24 Feb 12 2020 16:17:43

%S 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,

%T 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,

%U 14,14,15,15,16,16,15,15,16,17,17,16,16,15,15,14

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

%C This sequence is the real part of {f(n)} defined as:

%C - f(0) = 0,

%C - f(n+1) = f(n) + i^t(n)

%C where t(n) is the number of 1's and 6's minus the number of 3's and 4's

%C in the base 8 representation of n

%C and i denotes the imaginary unit.

%C We can also build the curve by successively applying the following substitution to an initial vector (1, 0):

%C .--->.

%C ^ |

%C | v

%C .--->. . .--->.

%C | ^

%C v |

%C .--->.

%H Rémy Sigrist, <a href="/A332246/b332246.txt">Table of n, a(n) for n = 0..4096</a>

%H Robert Ferréol (MathCurve), <a href="https://www.mathcurve.com/fractals/minkowski/minkowski.shtml">Saucisse de Minkowski</a> [in French]

%H Rémy Sigrist, <a href="/A332246/a332246.png">Representation of the first 1+8^4 points of the Minkowski sausage</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Minkowski_Sausage">Minkowski sausage</a>

%H <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>

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

%o (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)))) }

%Y See A163528, A323258 and A332204 for similar sequences.

%Y Cf. A332247 (Y-coordinates).

%K nonn,base

%O 0,4

%A _Rémy Sigrist_, Feb 08 2020