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

%I A176416 #29 Jun 04 2025 10:53:01
%S A176416 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,
%T A176416 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,
%U A176416 -1,-1,0,1,0,1,1,-1,-1,0,1,0,1,1,-1,-1,0,-1,0,1
%N A176416 Fixed point of morphism 0->0PPMM00, P->0PPMM0P, M=0PPMM0M (where P=+1, M=-1).
%C A176416 Turns by 120 degrees of a dragon curve (see fxtbook link below).
%C A176416 Also fixed point of morphism F->F0FPFPFMFMF0F, 0->0, P->P, M->M (after deleting all F).
%C A176416 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].
%C A176416 This is a 7-automatic sequence. - _Joerg Arndt_, Nov 09 2023
%H A176416 Paolo Xausa, <a href="/A176416/b176416.txt">Table of n, a(n) for n = 0..10000</a>
%H A176416 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 1.31.5 "Dragon curves based on radix-R counting", pp.95-101; image on p.98
%t A176416 First[SubstitutionSystem[{t_ :> {0, 1, 1, -1, -1, 0, t}}, {0}, {3}]] (* _Paolo Xausa_, Jun 04 2025 *)
%o A176416 (C++) /* CAT-algorithm */
%o A176416 int bit_dragon_r7_2_turn(unsigned long &x)
%o A176416 /* Increment the radix-7 word x and return (tr)
%o A176416    according to the lowest nonzero digit d of the incremented word:
%o A176416    d==[1,2,3,4,5,6] ==> rt:=[0,+1,+1,-1,-1,0] */
%o A176416 {
%o A176416   unsigned long s = 0;
%o A176416   while ( (x & 7) == 6 ) { x >>= 3; ++s; } /* scan over nines */
%o A176416   ++x; /* increment next digit */
%o A176416   int tr = 2 - ( (0x2f58 >> (2*(x&7)) ) & 3 ); x <<= (3*s); /* shift back */
%o A176416   return tr;
%o A176416 }
%Y A176416 Cf. A080846 (with terdragon curve), A014577 (with Heighway dragon), A175337 (with R5-dragon), and A176405 (with R7-dragon).
%K A176416 sign
%O A176416 0,1
%A A176416 _Joerg Arndt_, Apr 17 2010