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%I A332249 #22 Feb 12 2020 16:18:06
%S A332249 0,1,1,2,2,3,3,2,2,3,3,4,4,5,5,6,6,7,7,6,6,7,7,8,8,9,9,8,8,9,9,8,8,7,
%T A332249 7,6,6,5,5,6,6,7,7,8,8,9,9,8,8,9,9,10,10,11,11,12,12,11,11,12,12,13,
%U A332249 13,14,14,15,15,16,16,15,15,16,16,17,17,18,18,19
%N A332249 a(n) is the X-coordinate of the n-th point of the quadratic Koch curve. Sequence A332250 gives Y-coordinates.
%C A332249 This sequence is the real part of {f(n)} defined as:
%C A332249 - f(0) = 0,
%C A332249 - f(n+1) = f(n) + i^t(n)
%C A332249   where t(n) is the number of 1's minus the number of 3's
%C A332249              in the base 5 representation of n
%C A332249     and i denotes the imaginary unit.
%C A332249 We can also build the curve by successively applying the following substitution to an initial vector (1, 0):
%C A332249                         .--->.
%C A332249                         ^    |
%C A332249                         |    v
%C A332249                    .--->.    .--->.
%H A332249 Rémy Sigrist, <a href="/A332249/b332249.txt">Table of n, a(n) for n = 0..15625</a>
%H A332249 Robert Ferréol (MathCurve), <a href="https://www.mathcurve.com/fractals/kochquadratique/kochquadratique.shtml">Courbe de Koch quadratique</a> [in French]
%H A332249 Rémy Sigrist, <a href="/A332249/a332249.png">Representation of the first 1+2^10 points of the quadratic Koch curve</a>
%H A332249 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%F A332249 a(5^k-m) + a(m) = 3^k for any k >= 0 and m = 0..5^k.
%o A332249 (PARI) { k = [0, 1, 0, -1, 0]; z=0; for (n=0, 77, print1 (real(z) ", "); z += I^vecsum(apply(d -> k[1+d], digits(n, #k)))) }
%Y A332249 See A332246 for a similar sequence.
%Y A332249 Cf. A229217, A332250 (Y-coordinates).
%K A332249 nonn,base
%O A332249 0,4
%A A332249 _Rémy Sigrist_, Feb 08 2020

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