A039963 The period-doubling sequence A035263 repeated.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

An example of a d-perfect sequence.

Motzkin numbers mod 2. - Benoit Cloitre, Mar 23 2004

Let {a, b, c, c, a, b, a, b, a, b, c, c, a, b, ...} be the fixed point of the morphism: a -> ab, b -> cc, c -> ab, starting from a; then the sequence is obtained by taking a = 1, b = 1, c = 0. - Philippe Deléham, Mar 28 2004

The asymptotic mean of this sequence is 2/3 (Rowland and Yassawi, 2015; Burns, 2016). - Amiram Eldar, Jan 30 2021

The Gilbreath transform of floor(log_2(n)) (A000523). - Thomas Scheuerle, Sep 02 2024

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• T. Amdeberhan and M. Alekseyev, A moment sequence and Motzkin numbers. Modular coincidence?, MathOverflow, 2021.
• Rob Burns, Asymptotic density of Motzkin numbers modulo small primes, arXiv:1611.04910 [math.NT], 2016.
• David Kohel, San Ling and Chaoping Xing, Explicit Sequence Expansions, in: C. Ding, T. Helleseth and H. Niederreiter (eds.), Sequences and their Applications, Proceedings of SETA'98 (Singapore, 1998), Discrete Mathematics and Theoretical Computer Science, 1999, pp. 308-317; alternative link.
• Eric Rowland and Reem Yassawi, Automatic congruences for diagonals of rational functions, Journal de Théorie des Nombres de Bordeaux, Vol. 27, No. 1 (2015), pp. 245-288; arXiv preprint, arXiv:1310.8635 [math.NT], 2013-2014.

Crossrefs

Cf. A000523, A005802, A081706.

Motzkin numbers A001006 read mod 2,3,4,5,6,7,8,11: A039963, A039964, A299919, A258712, A299920, A258711, A299918, A258710.

Programs

Mathematica

Flatten[ Nest[ Function[l, {Flatten[(l /. {a -> {a, b}, b -> {c, c}, c -> {a, b}})]}], {a}, 7] /. {a -> {1}, b -> {1}, c -> {0}}] (* Robert G. Wilson v, Feb 26 2005 *)

PARI

A039963(n) = 1 - valuation(n\2+1,2)%2; \\ Max Alekseyev, Oct 23 2021

Python

def A039963(n): return ((m:=(n>>1)+1)&-m).bit_length()&1 # Chai Wah Wu, Jan 09 2023

Formula

a(n) = A035263(1+floor(n/2)). - Benoit Cloitre, Mar 23 2004

a(n) = A040039(n) mod 2 = A002212(n+1) mod 2. a(0) = a(1) = 1, for n>=2: a(n) = ( a(n) + Sum_{k=0..n-2} a(k)*a(n-2-k)) mod 2. - Philippe Deléham, Mar 26 2004

a(n) = (A(n+2) - A(n)) mod 2, for A = A019300, A001285, A010060, A010059, A000069, A001969. - Philippe Deléham, Mar 28 2004

a(n) = A001006(n) mod 2. - Christian G. Bower, Jun 12 2005

a(n) = (-1)^n*(A096268(n+1) - A096268(n)). - Johannes W. Meijer, Feb 02 2013

a(n) = 1 - A007814(floor(n/2)+1) mod 2 = A005802(n) mod 2. - Max Alekseyev, Oct 23 2021

Extensions

More terms from Christian G. Bower, Jun 12 2005

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe and Ralf Stephan, Jul 13 2007