A014707 a(4n) = 0, a(4n+2) = 1, a(2n+1) = a(n).

0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0

Offset

0,1

Comments

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

It appears that the sequence of run lengths is A088431. - Dimitri Hendriks, May 06 2010

Runs of three consecutive ones appear around positions n = 22, 46, 54, 86, 94, 118, 150, 174, 182, ..., or for n of the form 2^(k+3)*(4*t+3)-2, k >= 0, t >= 0. - Vladimir Shevelev, Mar 19 2011

References

G. Melançon, Factorizing infinite words using Maple, MapleTech journal, vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

J.-P. Allouche, M. Mendes France, A. Lubiw, A. J. van der Poorten and J. Shallit, Convergents of folded continued fractions, Acta Arithmetica 77 (1996), 77-96.

Cristina Ballantine and George Beck, Partitions enumerated by self-similar sequences, arXiv:2303.11493 [math.CO], 2023.

Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.

G. J. Endrullis, D. Hendriks and J. W. Klop, Degrees of streams, see table 1 "PF".

J.-Y. Kao et al., Words avoiding repetitions in arithmetic progressions

G. Melançon, Lyndon factorization of infinite words, STACS 96 (Grenoble, 1996), 147-154, Lecture Notes in Comput. Sci., 1046, Springer, Berlin, 1996. Math. Rev. 98h:68188.

Index entries for sequences obtained by enumerating foldings

Formula

a(A091072(n)-1) = 0; a(A091067(n)-1) = 1. - Reinhard Zumkeller, Sep 28 2011 [Adjusted to match offset by Peter Munn, Jul 01 2022]

a(n) = (1-Jacobi(-1,n+1))/2 (cf. A034947). - N. J. A. Sloane, Jul 27 2012 [Adjusted to match offset by Peter Munn, Jul 01 2022]

Set a=0, 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((2*n+1)*2^p-1) = n mod 2, p >= 0. - Johannes W. Meijer, Jan 28 2013

Maple

nmax:=92: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 0 to ceil(nmax/(p+2))+1 do a((2*n+1)*2^p-1) := n mod 2 od: od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Jan 28 2013
# second Maple program:
a:= proc(n) option remember;
     `if`(n::even, irem(n/2, 2), a((n-1)/2))
    end:
seq(a(n), n=0..92);  # Alois P. Heinz, Jun 27 2022

Mathematica

a[n_ /; Mod[n, 4] == 0] = 0; a[n_ /; Mod[n, 4] == 2] = 1; a[n_ /; Mod[n, 2] == 1] := a[n] = a[(n - 1)/2]; Table[a[n], {n, 0, 92}] (* Jean-François Alcover, May 17 2011 *)

Prog

(Haskell)
a014707 n = a014707_list !! n
a014707_list = f 0 $ cycle [0, 0, 1, 0] where
   f i (x:_:xs) = x : a014707 i : f (i+1) xs
-- Reinhard Zumkeller, Sep 28 2011
(Python)
def A014707(n):
    s = bin(n+1)[2:]
    m = len(s)
    i = s[::-1].find('1')
    return int(s[m-i-2]) if m-i-2 >= 0 else 0 # Chai Wah Wu, Apr 08 2021
(PARI) a(n)=n+=1; my(h=bitand(n, -n)); n=bitand(n, h<<1); n!=0; \\ Joerg Arndt, Apr 09 2021

Crossrefs

Equals 1 - A014577, which see for further references. Also a(n) = A038189(n+1).

The following are all essentially the same sequence: A014577, A014707, A014709, A014710, A034947, A038189, A082410.

Cf. A220466

Sequence in context: A131378 A354029 A189624 * A288213 A308187 A289007

Adjacent sequences: A014704 A014705 A014706 * A014708 A014709 A014710

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Scott C. Lindhurst (ScottL(AT)alumni.princeton.edu)

Status

approved