A014710 The regular paper-folding (or dragon curve) sequence. Alphabet {2,1}.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..10000
• G. Melançon, Factorizing infinite words using Maple, MapleTech journal, vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.
• 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

Programs

Mathematica

Nest[Append[#1, If[EvenQ[#2], 2 - BitAnd[1, #2/2], #1[[Ceiling[#2/2]]]]] & @@ {#, Length@ #} &, {2}, 105] (* Michael De Vlieger, Apr 08 2021 *)

PARI

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

Python

def A014710(n):
    s = bin(n+1)[2:]
    m = len(s)
    i = s[::-1].find('1')
    return 2-int(s[m-i-2]) if m-i-2 >= 0 else 2 # Chai Wah Wu, Apr 08 2021

Formula

Set a=2, b=1, 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(4*n) = 2, a(4*n+2) = 1, a(2*n+1) = a(n).