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a(n) is the X-coordinate of the n-th point of a variant of the quadratic Koch curve. Sequence A340328 gives Y-coordinates.
3

%I #12 Jan 07 2021 10:31:31

%S 0,1,1,2,2,3,3,2,2,3,3,4,4,5,5,4,4,5,5,6,6,7,7,6,6,7,7,6,6,5,5,4,4,5,

%T 5,6,6,7,7,6,6,7,7,8,8,9,9,8,8,9,9,10,10,11,11,10,10,9,9,8,8,9,9,8,8,

%U 9,9,10,10,11,11,10,10,11,11,12,12,13,13,12

%N a(n) is the X-coordinate of the n-th point of a variant of the quadratic Koch curve. Sequence A340328 gives Y-coordinates.

%C The curve is built by successively applying the following substitution to an initial vector (1, 0) (we have 4 copies and a horizontal unit vector):

%C .-.

%C ^ |

%C | |

%C | v

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

%C The curve visits once or twice every lattice point (x, y) such that 0 <= y <= x.

%C The quadratic Koch curve is built from 5 copies at each step.

%H Rémy Sigrist, <a href="/A340327/b340327.txt">Table of n, a(n) for n = 0..5461</a>

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

%H Rémy Sigrist, <a href="/A340327/a340327.png">Line plot of the first 5462 points</a>

%H Rémy Sigrist, <a href="/A340327/a340327.gp.txt">PARI program for A340327</a>

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

%e The curve starts as follows:

%e +

%e |42

%e |

%e +----+

%e |40 41

%e |

%e +----+----+

%e |34 |35 |38

%e | |39 |

%e +----+ +----+

%e |32 33 36 37

%e |

%e +----+----+ +----+

%e |10 |11 |30 |27 |26

%e | |31 | | |

%e +----+ +----+----+----+

%e |8 9 12 |13 |24 25

%e | |29 |28

%e +----+----+ +----+----+----+

%e |2 |3 |6 |15 |14 |19 |22

%e | |7 | | |18 |23 |

%e +----+ +----+ +----+ +----+

%e 0 1 4 5 16 17 20 21

%e - so a(0) = 0,

%e a(5) = a(6) = a(9) = a(10) = 3.

%o (PARI) See Links section.

%Y See A332249 and A340320 for similar sequences.

%Y Cf. A340328 (Y-coordinates).

%K nonn

%O 0,4

%A _Rémy Sigrist_, Jan 04 2021