cp's OEIS Frontend

A332383 a(n) is the X-coordinate of the n-th point of the dragon curve. Sequence A332384 gives Y-coordinates.

%I A332383 #24 Feb 16 2025 08:33:59
%S A332383 0,1,1,0,0,-1,-1,-2,-2,-3,-3,-2,-2,-3,-3,-4,-4,-5,-5,-4,-4,-3,-3,-2,
%T A332383 -2,-3,-3,-2,-2,-3,-3,-4,-4,-5,-5,-4,-4,-3,-3,-2,-2,-1,-1,-2,-2,-1,-1,
%U A332383 0,0,-1,-1,0,0,1,1,2,2,1,1,2,2,1,1,0,0,-1,-1,0,0,1,1,2
%N A332383 a(n) is the X-coordinate of the n-th point of the dragon curve. Sequence A332384 gives Y-coordinates.
%C A332383 To build the curve:
%C A332383 - start from the origin looking to the right,
%C A332383 - for k = 0, 1, ...:
%C A332383     - move forward to the next lattice point,
%C A332383     - if A014577(n) = 1 then turn 90 degrees to the left
%C A332383       otherwise turn 90 degrees to the right.
%H A332383 Rémy Sigrist, <a href="/A332383/b332383.txt">Table of n, a(n) for n = 0..8192</a>
%H A332383 Rémy Sigrist, <a href="/A332383/a332383.png">Colored representation of the first 2^18 points</a>
%H A332383 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DragonCurve.html">Dragon Curve</a>
%H A332383 Wikipedia, <a href="https://en.wikipedia.org/wiki/Dragon_curve">Dragon curve</a>
%H A332383 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%F A332383 For any k >= 0:
%F A332383 - a(2^(4*k))   =    (-4)^k,
%F A332383 - a(2^(4*k+1)) =    (-4)^k,
%F A332383 - a(2^(4*k+2)) =         0,
%F A332383 - a(2^(4*k+3)) = -2*(-4)^k.
%t A332383 Re[Join[{0}, Accumulate[Nest[Join[#, Reverse[I #]] &, {1}, 7]]]] (* _Vladimir Reshetnikov_, Apr 14 2022 *)
%o A332383 (PARI) A014577(n)=1/2*(1+(-1)^(1/2*((n+1)/2^valuation(n+1, 2)-1)))
%o A332383 { z=0; d=1; for (n=0, 71, print1 (real(z) ", "); z += d; d*=if (A014577(n), +I, -I)) }
%Y A332383 See A332251 for a similar sequence.
%Y A332383 Cf. A014577, A332384 (Y-coordinates).
%K A332383 sign,look,base
%O A332383 0,8
%A A332383 _Rémy Sigrist_, Feb 10 2020