A039964 Motzkin numbers A001006 read mod 3.

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

Offset

0,3

Comments

An example of a d-perfect sequence.

The asymptotic mean of this sequence is 0 (Burns, 2016). - Amiram Eldar, Jan 30 2021

Links

Amiram Eldar, Table of n, a(n) for n = 0..10000

Rob Burns, Asymptotic density of Motzkin numbers modulo small primes, arXiv:1611.04910 [math.NT], 2016.

Anders Hyllengren, Letter to N. J. A. Sloane, Oct 04 1985.

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.

Formula

a(n) = A001006(n) mod 3. - Christian G. Bower, Jun 12 2005

Mathematica

b = DifferenceRoot[Function[{b, n}, {3 (n + 1) b[n] + (2 n + 5) b[n + 1] == (n + 4) b[n + 2], b[0] == 1, b[1] == 1}]];
a[n_] := Mod[b[n], 3];
Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Feb 26 2019 *)

Prog

(PARI) a001006(n) = polcoeff((1-x-sqrt((1-x)^2-4*x^2+x^3*O(x^n)))/(2*x^2), n);
vector(200, n, n--; a001006(n) % 3) \\ Altug Alkan, Oct 23 2015

Crossrefs

Cf. A001006.

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

Sequence in context: A026821 A377334 A376971 * A369453 A340655 A035172

Adjacent sequences: A039961 A039962 A039963 * A039965 A039966 A039967

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Christian G. Bower, Jun 12 2005

Offset adapted by Altug Alkan, Oct 23 2015

Status

approved