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%I #41 Jan 30 2021 04:26:06
%S 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,
%T 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,
%U 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
%N Motzkin numbers A001006 read mod 3.
%C An example of a d-perfect sequence.
%C The asymptotic mean of this sequence is 0 (Burns, 2016). - _Amiram Eldar_, Jan 30 2021
%H Amiram Eldar, <a href="/A039964/b039964.txt">Table of n, a(n) for n = 0..10000</a>
%H 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 Anders Hyllengren, <a href="/A258710/a258710.pdf">Letter to N. J. A. Sloane, Oct 04 1985</a>.
%H 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>.
%F a(n) = A001006(n) mod 3. - _Christian G. Bower_, Jun 12 2005
%t 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}]];
%t a[n_] := Mod[b[n], 3];
%t Table[a[n], {n, 0, 104}] (* _Jean-François Alcover_, Feb 26 2019 *)
%o (PARI) a001006(n) = polcoeff((1-x-sqrt((1-x)^2-4*x^2+x^3*O(x^n)))/(2*x^2), n);
%o vector(200, n, n--; a001006(n) % 3) \\ _Altug Alkan_, Oct 23 2015
%Y Cf. A001006.
%Y Motzkin numbers A001006 read mod 2,3,4,5,6,7,8,11: A039963, A039964, A299919, A258712, A299920, A258711, A299918, A258710.
%K nonn
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Christian G. Bower_, Jun 12 2005
%E Offset adapted by _Altug Alkan_, Oct 23 2015
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