A332384 -id:A332384 - 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.

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

Rémy Sigrist, Feb 10 2020

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.

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

See A332251 for a similar sequence.

Cf. A014577, A332384 (Y-coordinates).

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

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)) }

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.

A355459 Real part of the Heighway/harter dragon curve points which are on the real axis.

Original entry on oeis.org

0, 1, -2, -3, -4, -5, 6, 7, 8, 7, 10, 11, 12, 13, 18, 17, 16, 15, 18, 19, 20, 21, -22, -23, -24, -23, -26, -27, -28, -29, -34, -33, -32, -33, -30, -29, -28, -27, -38, -39, -40, -39, -42, -43, -44, -45, -50, -49, -48, -47

Offset: 0

Reed Michael Upson, Jul 02 2022

sign

This sequence gives the values A332383(k) when A332384(k) = 0. - Rémy Sigrist, Oct 04 2022

Rémy Sigrist, PARI program

Cf. A246960, A332383, A332384, A355460.

PARI
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See Links section.
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A355460 Imaginary part of the Heighway/Harter dragon curve points which are on the imaginary axis.

Original entry on oeis.org

0, 1, 2, -3, -4, -5, -6, -9, -8, -9, -10, 11, 12, 13, 14, 17, 16, 15, 14, 19, 20, 21, 22, 25, 24, 25, 26, 37, 36, 35, 34, 31, 32, 31, 30, 35, 36, 37, 38, 41, 40, 41, 42, -43, -44, -45, -46, -49, -48, -47

Offset: 0

Reed Michael Upson, Jul 02 2022

sign

This sequence gives the values A332384(k) when A332383(k) = 0. - Rémy Sigrist, Oct 04 2022

Rémy Sigrist, PARI program

Cf. A246960, A332383, A332384, A355459.

PARI
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See Links section.
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A343949 Shortest distance from curve start to end along the segments of dragon curve expansion level n, and which is the diameter of the curve as a graph.

Original entry on oeis.org

1, 2, 4, 8, 12, 18, 26, 36, 52, 70, 102, 136, 200, 266, 394, 524, 780, 1038, 1550, 2064, 3088, 4114, 6162, 8212, 12308, 16406, 24598, 32792, 49176, 65562, 98330, 131100, 196636, 262174, 393246, 524320, 786464, 1048610, 1572898, 2097188, 3145764, 4194342, 6291494

Offset: 0

Kevin Ryde, May 05 2021

Expansion level n is the first 2^n segments of the curve, and can be taken as a graph with visited points as vertices and segments as edges.

			Curve n=4:
     *--*  *--*
     |  |  |  |        Start S to end E along segments.
     *--*--*  *--*     Distance a(4) = 12,
        |        |     which is also graph diameter.
  E  *--*     S--*
  |  |
  *--*

Kevin Ryde, Table of n, a(n) for n = 0..1000
Kevin Ryde, Iterations of the Dragon Curve, see index "Diameter".
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-2,2).

Cf. A275970, A083706, A228693.

Cf. A332383, A332384 (curve coordinates).

a(n) = if(n==0,1, my(t=n%2); (3+t)<<(n>>1) + n-4 + t);

a(0) = 1.
a(2*n)   = 3*2^n + 2*n - 4 = 2*A275970(n-1), for n>=1.
a(2*n+1) = 4*2^n + 2*n - 2 = 2*A083706(n).
a(n+1) - a(n) = 2*A228693(n), for n>=1.
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 2*a(n-4) + 2*a(n-5) for n >= 6.
G.f.: (1 + x - x^2 + x^3 - 4*x^5) / ((1+x) * (1-x)^2 * (1-2*x^2)).
G.f.: 2 - (1/2)/(1+x) - (9/2)/(1-x) + 1/(1-x)^2  + (3 + 4*x)/(1 - 2*x^2).

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.

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A355459 Real part of the Heighway/harter dragon curve points which are on the real axis.

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PARI

A355460 Imaginary part of the Heighway/Harter dragon curve points which are on the imaginary axis.

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PARI

A343949 Shortest distance from curve start to end along the segments of dragon curve expansion level n, and which is the diameter of the curve as a graph.

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Crossrefs

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PARI

Formula