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a(n) is the Y-coordinate of the n-th point of the dragon curve. Sequence A332383 gives X-coordinates.
5

%I #18 Apr 15 2022 13:02:22

%S 0,0,1,1,2,2,1,1,2,2,1,1,0,0,-1,-1,0,0,-1,-1,-2,-2,-1,-1,-2,-2,-3,-3,

%T -4,-4,-5,-5,-4,-4,-5,-5,-6,-6,-5,-5,-6,-6,-5,-5,-4,-4,-3,-3,-4,-4,-5,

%U -5,-6,-6,-5,-5,-6,-6,-7,-7,-8,-8,-9,-9,-8,-8,-9,-9,-10

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

%H Rémy Sigrist, <a href="/A332384/b332384.txt">Table of n, a(n) for n = 0..8192</a>

%H <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>

%F For any k >= 0:

%F - a(2^(4*k)) = 0,

%F - a(2^(4*k+1)) = (-4)^k,

%F - a(2^(4*k+2)) = 2*(-4)^k,

%F - a(2^(4*k+3)) = 2*(-4)^k.

%t Im[Join[{0}, Accumulate[Nest[Join[#, Reverse[I #]] &, {1}, 7]]]] (* _Vladimir Reshetnikov_, Apr 14 2022 *)

%o (PARI) A014577(n)=1/2*(1+(-1)^(1/2*((n+1)/2^valuation(n+1, 2)-1)))

%o { z=0; d=1; for (n=0, 68, print1 (imag(z) ", "); z += d; d*=if (A014577(n), +I, -I)) }

%Y Cf. A014577, A332383 (X-coordinates and additional comments).

%K sign,base

%O 0,5

%A _Rémy Sigrist_, Feb 10 2020