A039969 An example of a d-perfect sequence: a(n) = Catalan(n) mod 3.

1, 1, 2, 2, 2, 0, 0, 0, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 1, 1, 1, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 1, 1, 1, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

0,3

Comments

This is A006996 with all its terms repeated three times, except the initial term only twice. A006996 is a fixed point of the morphism 0 -> 000, 1 -> 120, 2 -> 210. [The original comment edited by Antti Karttunen, Aug 14 2017]

Equals Catalan(n) mod 3. (Cf. A000108.) - Paul D. Hanna, Jun 20 2003 [confirmed by Christian G. Bower, Jun 12 2005]

Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!).

Links

Antti Karttunen, Table of n, a(n) for n = 0..10000

Rob Burns, Asymptotic density of Catalan numbers modulo 3 and powers of 2, arXiv:1611.03705 [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.

Index entries for sequences that are fixed points of mappings

Formula

a(n) = ((-1)^(n+1)*A001006(n-1)) mod 3, for n>0. - Christian G. Bower, Jun 12 2005

a(n) = a(n-1) if n == 0 or 1 (mod 3). a(n) = 0 if n == 5,6, or 7 (mod 9). - Robert Israel, Sep 20 2015

a(3n) = A006996(n). - Antti Karttunen, Aug 14 2017

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = 0 (Burns, 2016). - Amiram Eldar, Jan 26 2021

Maple

seq(binomial(2*n, n)/(n+1) mod 3, n = 0 .. 100); # Robert Israel, Sep 20 2015

Mathematica

Take[ Flatten[ Nest[ Flatten[ # /. {1 -> {1, 2, 0}, 2 -> {2, 1, 0}, 0 -> {0, 0, 0}}] &, {1}, 4] /. {1 -> {1, 1, 1}, 2 -> {2, 2, 2}, 0 -> {0, 0, 0}}], {2, 106}] (* or *)
Table[ Mod[ Binomial[ 2n, n]/(n + 1), 3], {n, 0, 104}] (* Robert G. Wilson v, Sep 09 2005 *)
Mod[CatalanNumber[Range[0, 110]], 3] (* Harvey P. Dale, Oct 23 2017 *)

Prog

(Magma) [Catalan(n) mod 3: n in [1..80]]; // Vincenzo Librandi, Jul 14 2015
(PARI) A039969(n) = ((binomial(2*n, n)/(n+1))%3); \\ Antti Karttunen, Aug 13 2017

Crossrefs

Cf. A000108, A001006, A010872, A039972, A067397.

Cf. A006996 (trisection).

Sequence in context: A028953 A348648 A037865 * A039967 A258133 A327152

Adjacent sequences: A039966 A039967 A039968 * A039970 A039971 A039972

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Christian G. Bower, Jun 12 2005

Offset corrected from 1 to 0 by Antti Karttunen, Aug 13 2017

Status

approved