A014709 The regular paper-folding (or dragon curve) sequence.

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

0,3

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

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

Mathematica

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

Prog

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

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.

Sequence in context: A079882 A362415 A317335 * A278161 A069258 A273134

Adjacent sequences: A014706 A014707 A014708 * A014710 A014711 A014712

Keyword

nonn

Author

N. J. A. Sloane.

Status

approved