A006997 Partitioning integers to avoid arithmetic progressions of length 3.

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, 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, 5, 6, 6, 7, 4, 5, 5, 4, 5, 5, 6, 6, 7, 0, 0, 1, 0, 0, 1

Offset: 0

Author

N. J. A. Sloane, James Propp

Keywords

• nonn
• easy

Comments

a(n) = 0 iff n in A005836.

References

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

Links

• Ben Chen, Richard Chen, Joshua Guo, Tanya Khovanova, Shane Lee, Neil Malur, Nastia Polina, Poonam Sahoo, Anuj Sakarda, Nathan Sheffield, and Armaan Tipirneni, On Base 3/2 and its sequences, arXiv:1808.04304 [math.NT], 2018.
• Joseph Gerver, James Propp and Jamie Simpson, Greedily partitioning the natural numbers into sets free of arithmetic progressions Proc. Amer. Math. Soc. 102 (1988), no. 3, 765-772.
• A. M. Odlyzko and R. P. Stanley, Some curious sequences constructed with the greedy algorithm, 1978.
• James Propp and N. J. A. Sloane, Email, March 1994
• J. Shallit, k-regular Sequences
• J. Shallit, Number theory and formal languages, 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.

Crossrefs

Cf. A005836, A100480.

Formula

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

a(n) = A100480(n+1) - 1. - Pontus von Brömssen, Apr 09 2025