OFFSET
1
COMMENTS
A cubefree word.
A generalized choral sequence c(3n+r_0)=0, c(3n+r_1)=1, c(3n+r_c)=c(n), with r_0=0, r_1=1, and r_c=2. - Joel Reyes Noche (joel.noche(AT)up.edu.ph), Jul 09 2009
From Joerg Arndt, Apr 15 2010: (Start)
Turns (by 120 degrees) of the terdragon curve 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).
See fxtbook link below.
(End)
REFERENCES
J. Berstel and J. Karhumaki, Combinatorics on words - a tutorial, Bull. EATCS, #79 (2003), pp. 178-228.
J. R. Noche, Generalized Choral Sequences, Matimyas Matematika, 31(2008), 25-28. [From Joel Reyes Noche (joel.noche(AT)up.edu.ph), Jul 09 2009]
LINKS
Joerg Arndt Matters Computational (The Fxtbook), section 1.31.4, pp. 92-95; dragon curve picture on p. 93.
Jean Berstel, Home Page
Dimitri Hendriks, Frits G. W. Dannenberg, Jorg Endrullis, Mark Dow and Jan Willem Klop, Arithmetic Self-Similarity of Infinite Sequences, arXiv preprint 1201.3786 [math.CO], 2012.
Mazen Khodier, New Methods for Analyzing the Properties of Automatic Sequences, Master's Thesis, Univ. Waterloo (Canada 2026). See p. 27, Table 5.1.
Kaisei Kishi, Peaker Guo, Cristian Urbina, and Hideo Bannai, On Occurrence-Preserving Morphisms, arXiv:2605.18034 [math.CO], 2026. See p. 6 (Cor. 16).
FORMULA
a(n) = (A062756(n) - A062756(n+1) + 1)/2, where A062756(n) is the number of 1's in the ternary expansion of n. From formula in A062756: g.f.: A(x) = 1/(1-x)/2 - Sum_{k>=0} x^(3^k-1)/(1+x^(3^k)+x^(2*3^k))/2. - Paul D. Hanna, Feb 24 2006
Given g.f. A(x) then B(x) = x * A(x) satisfies B(x) = x^2 / (1 - x^3) + B(x^3). - Michael Somos, Jul 29 2009
a(3*n) = 0, a(3*n + 1) = 1, a(3*n - 1) = a(n - 1). - Michael Somos, Jul 29 2009
a(n) = -1 + A060236(n). - Joerg Arndt, Jan 21 2013
EXAMPLE
Start: 0
Rules:
0 --> 010
1 --> 011
-------------
0: (#=1)
0
1: (#=3)
010
2: (#=9)
010011010
3: (#=27)
010011010010011011010011010
4: (#=81)
010011010010011011010011010010011010010011011010011011010011010010011011010011010
MAPLE
a:= proc(n) option remember;
(r-> `if`(r<2, r, a(m+1)))(irem(n-1, 3, 'm'))
end:
seq(a(n), n=1..105); # Alois P. Heinz, Feb 04 2011
MATHEMATICA
Nest[Flatten[ # /. {0 -> {0, 1, 0}, 1 -> {0, 1, 1}}] &, {0}, 5]
PROG
(PARI) {a(n)=if(n<1, 0, polcoeff(1/(1-x)/2-sum(k=0, ceil(log(n+1)/log(3)), x^(3^k-1)/(1+x^(3^k)+x^(2*3^k)+x*O(x^n)))/2, n))} \\ Paul D. Hanna, Feb 24 2006
(PARI) {a(n) = if( n<1, 0, n++; n / 3^valuation(n, 3) % 3 -1 )} /* Michael Somos, Jul 29 2009 */
(C++) /* CAT algorithm */
bool bit_dragon3_turn(ulong &x)
/* Increment the radix-3 word x and return whether
the number of ones in x is decreased. */
{
ulong s = 0;
while ( (x & 3) == 2 ) { x >>= 2; ++s; } /* scan over nines */
bool tr = ( (x & 3) != 0 ); /* incremented word will have one less 1 */
++x; /* increment next digit */
x <<= (s<<1); /* shift back */
return tr;
} /* Joerg Arndt, Apr 15 2010 */
CROSSREFS
Cf. A189628 (guide).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Mar 29 2003
EXTENSIONS
More terms from Wouter Meeussen, Apr 01 2003
Offset changed from 0 to 1 by Michel Dekking, Oct 26 2019
STATUS
approved