A175337 Fixed point of morphism 0 -> 00110, 1 -> 00111

0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0

Offset

0,1

Comments

Turns (by 90 degrees) of a dragon curve (called R5-dragon in the fxtbook, see link below) which can be rendered as follows:

[Init] Set n=0 and direction=0.

[Draw] Draw a unit line (in the current direction). Turn left/right if a(n) is zero/nonzero respectively.

[Next] Set n=n+1 and goto (draw).

With counting in radix 5: whether the lowest nonzero digit is >2 (see C++ code).

With morphism F -> F0F0F1F1F, 0 -> 0, 1 -> 1: fixed point with all 'F' omitted.

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

Joerg Arndt, Matters Computational (The Fxtbook), section 1.31.5 "Dragon curves based on radix-R counting", pp.95-101; image on p.96

Mathematica

Nest[Flatten[ReplaceAll[#, {0->{0, 0, 1, 1, 0}, 1->{0, 0, 1, 1, 1}}]]&, {0}, 3] (* Paolo Xausa, Nov 09 2023 *)

Prog

(C++) /* CAT algorithm */
bool bit_dragon_r5_turn(ulong &x)
/* Increment the radix-5 word x and return (tr) whether
   the lowest nonzero digit of the incremented word is > 2. */
{
  ulong s = 0;
  while ( (x & 7) == 4 ) { x >>= 3; ++s; } /* scan over nines */
  bool tr = ( (x & 7) >= 2 ); /* whether digit will be > 2 */
  ++x; /* increment next digit */
  x <<= (3*s); /* shift back */
  return tr;
}

Crossrefs

Cf. A080846 (with terdragon curve) and A014577 (with Heighway dragon).

Sequence in context: A287382 A074290 A091225 * A285031 A327222 A286063

Adjacent sequences: A175334 A175335 A175336 * A175338 A175339 A175340

Keyword

nonn

Author

Joerg Arndt, Apr 15 2010

Status

approved