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A340328 a(n) is the Y-coordinate of the n-th point of a variant of the quadratic Koch curve. Sequence A340327 gives X-coordinates.

%I A340328 #11 Jan 07 2021 10:31:38
%S A340328 0,0,1,1,0,0,1,1,2,2,3,3,2,2,1,1,0,0,1,1,0,0,1,1,2,2,3,3,2,2,3,3,4,4,
%T A340328 5,5,4,4,5,5,6,6,7,7,6,6,5,5,4,4,5,5,4,4,3,3,2,2,3,3,2,2,1,1,0,0,1,1,
%U A340328 0,0,1,1,2,2,3,3,2,2,1,1,0,0,1,1,0,0,1
%N A340328 a(n) is the Y-coordinate of the n-th point of a variant of the quadratic Koch curve. Sequence A340327 gives X-coordinates.
%C A340328 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 A340328                            .-.
%C A340328                            ^ |
%C A340328                            | |
%C A340328                            | v
%C A340328                    .------>. .------>.
%C A340328 The curve visits once or twice every lattice point (x, y) such that 0 <= y <= x.
%C A340328 The quadratic Koch curve is built from 5 copies at each step.
%H A340328 Rémy Sigrist, <a href="/A340328/b340328.txt">Table of n, a(n) for n = 0..5461</a>
%H A340328 Robert Ferréol (MathCurve), <a href="https://www.mathcurve.com/fractals/kochquadratique/kochquadratique.shtml">Courbe de Koch quadratique</a> [in French]
%H A340328 Rémy Sigrist, <a href="/A340328/a340328.gp.txt">PARI program for A340328</a>
%H A340328 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%e A340328 The curve starts as follows:
%e A340328                                     +
%e A340328                                     |42
%e A340328                                     |
%e A340328                                +----+
%e A340328                                |40   41
%e A340328                                |
%e A340328                           +----+----+
%e A340328                           |34  |35  |38
%e A340328                           |    |39  |
%e A340328                      +----+    +----+
%e A340328                      |32   33   36   37
%e A340328                      |
%e A340328                 +----+----+    +----+
%e A340328                 |10  |11  |30  |27  |26
%e A340328                 |    |31  |    |    |
%e A340328            +----+    +----+----+----+
%e A340328            |8    9    12  |13  |24   25
%e A340328            |              |29  |28
%e A340328       +----+----+    +----+----+----+
%e A340328       |2   |3   |6   |15  |14  |19  |22
%e A340328       |    |7   |    |    |18  |23  |
%e A340328 +----+    +----+    +----+    +----+
%e A340328   0    1    4    5    16   17   20   21
%e A340328 - so a(8) = a(9) = a(12) = a(13) = a(24) = a(25) = a(28) = a(29) = 2,
%e A340328      a(40) = a(41) = 6.
%o A340328 (PARI) See Links section.
%Y A340328 See A332250 and A340321 for similar sequences.
%Y A340328 Cf. A340327 (X-coordinates).
%K A340328 nonn
%O A340328 0,9
%A A340328 _Rémy Sigrist_, Jan 04 2021