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

0, 1, 1, 0, 0, -1, -1, -2, -2, -3, -3, -2, -2, -3, -3, -4, -4, -5, -5, -4, -4, -3, -3, -2, -2, -3, -3, -2, -2, -3, -3, -4, -4, -5, -5, -4, -4, -3, -3, -2, -2, -1, -1, -2, -2, -1, -1, 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

Offset: 0

Author

Rémy Sigrist, Feb 10 2020

Keywords

• sign
• look
• base

Comments

To build the curve:

- start from the origin looking to the right,

- for k = 0, 1, ...:

- move forward to the next lattice point,

- if A014577(n) = 1 then turn 90 degrees to the left

otherwise turn 90 degrees to the right.

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8192
• Rémy Sigrist, Colored representation of the first 2^18 points
• Eric Weisstein's World of Mathematics, Dragon Curve
• Wikipedia, Dragon curve
• Index entries for sequences related to coordinates of 2D curves

Crossrefs

See A332251 for a similar sequence.

Cf. A014577, A332384 (Y-coordinates).

Programs

Mathematica

Re[Join[{0}, Accumulate[Nest[Join[#, Reverse[I #]] &, {1}, 7]]]] (* Vladimir Reshetnikov, Apr 14 2022 *)

PARI

A014577(n)=1/2*(1+(-1)^(1/2*((n+1)/2^valuation(n+1, 2)-1)))
{ z=0; d=1; for (n=0, 71, print1 (real(z) ", "); z += d; d*=if (A014577(n), +I, -I)) }

Formula

For any k >= 0:

- a(2^(4*k)) = (-4)^k,

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

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

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