A014709 The regular paper-folding (or dragon curve) sequence. Alphabet {1,2}.

1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Over the alphabet {a,b} this is aabaabbaaabbabbaaabaabbbaabbabbaaaba...

With offset 1, completely multiplicative modulo 3. - Peter Munn, Jun 20 2022

References

• J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, pp. 155, 182.
• G. Melançon, Factorizing infinite words using Maple, MapleTech journal, vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..10000
• Gabriele Fici and Luca Q. Zamboni, On the least number of palindromes contained in an infinite word, Theoretical Computer Science, Volume 481, 2013, pp. 1-8. See page 1.
• Index entries for sequences obtained by enumerating foldings

Crossrefs

See A014577 for more references and more terms.

The following are all essentially the same sequence: A014577, A014707, A014709, A014710, A034947, A038189, A082410. - N. J. A. Sloane, Jul 27 2012

Cf. A065339.

Programs

Mathematica

(3 - JacobiSymbol[-1, Range[100]])/2 (* Paolo Xausa, May 26 2024 *)

PARI

a(n)=if(n%2==0, 1+bitand(1,n\2), a(n\2) );
for(n=0,122,print1(a(n),", "))

Formula

Set a=1, b=2, S(0)=a, S(n+1) = S(n)aF(S(n)), where F(x) reverses x and then interchanges a and b; sequence is limit S(infinity).

a(4n) = 1, a(4n+2) = 2, a(2n+1) = a(n).

a(n) = (3-jacobi(-1,n+1))/2 (cf. A034947). - N. J. A. Sloane, Jul 27 2012 [index adjusted by Peter Munn, Jun 22 2022]

a(n) = 1 + A065339(n+1) mod 2. - Peter Munn, Jun 20 2022