cp's OEIS Frontend

A039963 The period-doubling sequence A035263 repeated.

%I A039963 #56 Sep 05 2024 14:13:47
%S A039963 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,
%T A039963 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,
%U A039963 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
%N A039963 The period-doubling sequence A035263 repeated.
%C A039963 An example of a d-perfect sequence.
%C A039963 Motzkin numbers mod 2. - _Benoit Cloitre_, Mar 23 2004
%C A039963 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
%C A039963 The asymptotic mean of this sequence is 2/3 (Rowland and Yassawi, 2015; Burns, 2016). - _Amiram Eldar_, Jan 30 2021
%C A039963 The Gilbreath transform of floor(log_2(n)) (A000523). - _Thomas Scheuerle_, Sep 02 2024
%H A039963 Seiichi Manyama, <a href="/A039963/b039963.txt">Table of n, a(n) for n = 0..10000</a>
%H A039963 T. Amdeberhan and M. Alekseyev, <a href="https://mathoverflow.net/q/406773">A moment sequence and Motzkin numbers. Modular coincidence?</a>, MathOverflow, 2021.
%H A039963 Rob Burns, <a href="https://arxiv.org/abs/1611.04910">Asymptotic density of Motzkin numbers modulo small primes</a>, arXiv:1611.04910 [math.NT], 2016.
%H A039963 David Kohel, San Ling and Chaoping Xing, <a href="https://doi.org/10.1007/978-1-4471-0551-0_23">Explicit Sequence Expansions</a>, 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; <a href="http://www.maths.usyd.edu.au/u/kohel/doc/perfect.ps">alternative link</a>.
%H A039963 Eric Rowland and Reem Yassawi, <a href="http://www.numdam.org/item/JTNB_2015__27_1_245_0/">Automatic congruences for diagonals of rational functions</a>, Journal de Théorie des Nombres de Bordeaux, Vol. 27, No. 1 (2015), pp. 245-288; <a href="http://arxiv.org/abs/1310.8635">arXiv preprint</a>, arXiv:1310.8635 [math.NT], 2013-2014.
%F A039963 a(n) = A035263(1+floor(n/2)). - _Benoit Cloitre_, Mar 23 2004
%F A039963 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
%F A039963 a(n) = (A(n+2) - A(n)) mod 2, for A = A019300, A001285, A010060, A010059, A000069, A001969. - _Philippe Deléham_, Mar 28 2004
%F A039963 a(n) = A001006(n) mod 2. - _Christian G. Bower_, Jun 12 2005
%F A039963 a(n) = (-1)^n*(A096268(n+1) - A096268(n)). - _Johannes W. Meijer_, Feb 02 2013
%F A039963 a(n) = 1 - A007814(floor(n/2)+1) mod 2 = A005802(n) mod 2. - _Max Alekseyev_, Oct 23 2021
%t A039963 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 *)
%o A039963 (PARI) A039963(n) = 1 - valuation(n\2+1,2)%2; \\ _Max Alekseyev_, Oct 23 2021
%o A039963 (Python)
%o A039963 def A039963(n): return ((m:=(n>>1)+1)&-m).bit_length()&1 # _Chai Wah Wu_, Jan 09 2023
%Y A039963 Cf. A000523, A005802, A081706.
%Y A039963 Motzkin numbers A001006 read mod 2,3,4,5,6,7,8,11: A039963, A039964, A299919, A258712, A299920, A258711, A299918, A258710.
%K A039963 nonn
%O A039963 0,1
%A A039963 _N. J. A. Sloane_
%E A039963 More terms from _Christian G. Bower_, Jun 12 2005
%E A039963 Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_ and _Ralf Stephan_, Jul 13 2007