A176416 Fixed point of morphism 0->0PPMM00, P->0PPMM0P, M=0PPMM0M (where P=+1, M=-1)

0, 1, 1, -1, -1, 0, 0, 0, 1, 1, -1, -1, 0, 1, 0, 1, 1, -1, -1, 0, 1, 0, 1, 1, -1, -1, 0, -1, 0, 1, 1, -1, -1, 0, -1, 0, 1, 1, -1, -1, 0, 0, 0, 1, 1, -1, -1, 0, 0, 0, 1, 1, -1, -1, 0, 0, 0, 1, 1, -1, -1, 0, 1, 0, 1, 1, -1, -1, 0, 1, 0, 1, 1, -1, -1, 0, -1, 0, 1

Offset

0,1

Comments

Turns by 120 degrees of a dragon curve (see fxtbook link below).

Also fixed point of morphism F->F0FPFPFMFMF0F, 0->0, P->P, M->M (after deleting all F).

Let d(n) be the lowest nonzero digit in the radix-7 expansion of (n+1), then if d(n)==[1,2,3,4,5,6] ==> a(n):=[0,+1,+1,-1,-1,0].

This is a 7-automatic sequence. - Joerg Arndt, Nov 09 2023

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.98

Mathematica

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

Prog

(C++) /* CAT-algorithm */
int bit_dragon_r7_2_turn(unsigned long &x)
/* Increment the radix-7 word x and return (tr)
   according to the lowest nonzero digit d of the incremented word:
   d==[1, 2, 3, 4, 5, 6] ==> rt:=[0, +1, +1, -1, -1, 0] */
{
  unsigned long s = 0;
  while ( (x & 7) == 6 ) { x >>= 3; ++s; } /* scan over nines */
  ++x; /* increment next digit */
  int tr = 2 - ( (0x2f58 >> (2*(x&7)) ) & 3 ); x <<= (3*s); /* shift back */
  return tr;
}

Crossrefs

Cf. A080846 (with terdragon curve), A014577 (with Heighway dragon), A175337 (with R5-dragon), and A176405 (with R7-dragon).

Sequence in context: A138886 A269528 A099859 * A102460 A080908 A131720

Adjacent sequences: A176413 A176414 A176415 * A176417 A176418 A176419

Keyword

sign

Author

Joerg Arndt, Apr 17 2010

Status

approved