cp's OEIS Frontend

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

%I A340327 #12 Jan 07 2021 10:31:31
%S A340327 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 A340327 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 A340327 9,9,10,10,11,11,10,10,11,11,12,12,13,13,12
%N A340327 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 A340327 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 A340327                            .-.
%C A340327                            ^ |
%C A340327                            | |
%C A340327                            | v
%C A340327                    .------>. .------>.
%C A340327 The curve visits once or twice every lattice point (x, y) such that 0 <= y <= x.
%C A340327 The quadratic Koch curve is built from 5 copies at each step.
%H A340327 Rémy Sigrist, <a href="/A340327/b340327.txt">Table of n, a(n) for n = 0..5461</a>
%H A340327 Robert Ferréol (MathCurve), <a href="https://www.mathcurve.com/fractals/kochquadratique/kochquadratique.shtml">Courbe de Koch quadratique</a> [in French]
%H A340327 Rémy Sigrist, <a href="/A340327/a340327.png">Line plot of the first 5462 points</a>
%H A340327 Rémy Sigrist, <a href="/A340327/a340327.gp.txt">PARI program for A340327</a>
%H A340327 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%e A340327 The curve starts as follows:
%e A340327                                     +
%e A340327                                     |42
%e A340327                                     |
%e A340327                                +----+
%e A340327                                |40   41
%e A340327                                |
%e A340327                           +----+----+
%e A340327                           |34  |35  |38
%e A340327                           |    |39  |
%e A340327                      +----+    +----+
%e A340327                      |32   33   36   37
%e A340327                      |
%e A340327                 +----+----+    +----+
%e A340327                 |10  |11  |30  |27  |26
%e A340327                 |    |31  |    |    |
%e A340327            +----+    +----+----+----+
%e A340327            |8    9    12  |13  |24   25
%e A340327            |              |29  |28
%e A340327       +----+----+    +----+----+----+
%e A340327       |2   |3   |6   |15  |14  |19  |22
%e A340327       |    |7   |    |    |18  |23  |
%e A340327 +----+    +----+    +----+    +----+
%e A340327   0    1    4    5    16   17   20   21
%e A340327 - so a(0) = 0,
%e A340327      a(5) = a(6) = a(9) = a(10) = 3.
%o A340327 (PARI) See Links section.
%Y A340327 See A332249 and A340320 for similar sequences.
%Y A340327 Cf. A340328 (Y-coordinates).
%K A340327 nonn
%O A340327 0,4
%A A340327 _Rémy Sigrist_, Jan 04 2021