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A340320 a(n) is the X-coordinate of the n-th point of a variant of the quadratic Koch curve. Sequence A340321 gives Y-coordinates.

%I A340320 #13 Jan 07 2021 10:31:17
%S A340320 0,1,1,2,2,3,3,4,4,3,3,4,4,5,5,6,6,5,5,6,6,7,7,8,8,9,9,10,10,9,9,10,
%T A340320 10,11,11,12,12,13,13,12,12,11,11,10,10,9,9,10,10,9,9,10,10,11,11,12,
%U A340320 12,13,13,12,12,13,13,14,14,15,15,14,14,15,15,16,16
%N A340320 a(n) is the X-coordinate of the n-th point of a variant of the quadratic Koch curve. Sequence A340321 gives Y-coordinates.
%C A340320 The curve is built by successively applying the following substitution to an initial vector (1, 0) (the two vertical copies are horizontally flipped):
%C A340320                                *
%C A340320                            .------>.
%C A340320                            ^       |
%C A340320                            |*     *|
%C A340320                        *   |       v   *
%C A340320                    .------>.       .------>.
%C A340320 The quadratic Koch curve is built without horizontal flip.
%H A340320 Rémy Sigrist, <a href="/A340320/b340320.txt">Table of n, a(n) for n = 0..3125</a>
%H A340320 Robert Ferréol (MathCurve), <a href="https://www.mathcurve.com/fractals/kochquadratique/kochquadratique.shtml">Courbe de Koch quadratique</a> [in French]
%H A340320 Rémy Sigrist, <a href="/A340320/a340320.png">Line plot of the first 1+5^5 points</a>
%H A340320 Rémy Sigrist, <a href="/A340320/a340320.gp.txt">PARI program for A340320</a>
%H A340320 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%e A340320 The curve starts as follows:
%e A340320                      +---+
%e A340320                      |12 |13
%e A340320                      |   |
%e A340320                  +---+   +---+
%e A340320                  |10  11  14 |15
%e A340320                  |           |
%e A340320                  +---+   +---+
%e A340320                   9  |8  |17  16
%e A340320                      |   |
%e A340320          +---+   +---+   +---+   +---+
%e A340320          |2  |3  |6   7   18 |19 |22 |23
%e A340320          |   |   |           |   |   |
%e A340320      +---+   +---+           +---+   +---+
%e A340320       0   1   4   5           20  21  24  25
%e A340320 - so a(0) = 0,
%e A340320      a(5) = a(6) = a(9) = a(10) = 3.
%o A340320 (PARI) See Links section.
%Y A340320 See A332249 and A340327 for similar sequences.
%Y A340320 Cf. A340321 (Y-coordinates).
%K A340320 nonn
%O A340320 0,4
%A A340320 _Rémy Sigrist_, Jan 04 2021