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Partitioning integers to avoid arithmetic progressions of length 3.
(Formerly M0185)
11

%I M0185 #39 Apr 20 2023 09:23:30

%S 0,0,1,0,0,1,1,2,2,0,0,1,0,0,1,1,2,2,1,2,2,3,3,4,3,3,4,0,0,1,0,0,1,1,

%T 2,2,0,0,1,0,0,1,1,2,2,1,2,2,3,3,4,3,3,4,1,2,2,3,3,4,3,3,4,4,5,5,4,5,

%U 5,6,6,7

%N Partitioning integers to avoid arithmetic progressions of length 3.

%C a(n) = 0 iff n in A005836.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H B. Chen, R. Chen, J. Guo, S. Lee et al, <a href="https://arxiv.org/abs/1808.04304">On Base 3/2 and its sequences</a>, arXiv:1808.04304 [math.NT], 2018.

%H Joseph Gerver, James Propp and Jamie Simpson, <a href="http://dx.doi.org/10.1090/S0002-9939-1988-0929018-1">Greedily partitioning the natural numbers into sets free of arithmetic progressions</a> Proc. Amer. Math. Soc. 102 (1988), no. 3, 765-772.

%H A. M. Odlyzko and R. P. Stanley, <a href="https://math.mit.edu/~rstan/papers/od.pdf">Some curious sequences constructed with the greedy algorithm</a>, 1978.

%H James Propp and N. J. A. Sloane, <a href="/A006997/a006997.pdf">Email, March 1994</a>

%H J. Shallit, <a href="https://cs.uwaterloo.ca/~shallit/Talks/kreg7.pdf">k-regular Sequences</a>

%H J. Shallit, <a href="https://cs.uwaterloo.ca/~shallit/Papers/ntfl.pdf">Number theory and formal languages</a>, in D. A. Hejhal, J. Friedman, M. C. Gutzwiller and A. M. Odlyzko, eds., Emerging Applications of Number Theory, IMA Volumes in Mathematics and Its Applications, V. 109, Springer-Verlag, 1999, pp. 547-570.

%F a(3n+k) = floor((3*a(n)+k)/2), 0 <= k <= 2.

%K nonn,easy

%O 0,8

%A _N. J. A. Sloane_, _James Propp_