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%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
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