Given two starting numbers , the following table gives the unique sequences {a_i}" src="https://mathworld.wolfram.com/images/equations/NonarithmeticProgressionSequence/Inline2.gif" style="height:15px; width:21px" /> that contain no three-term arithmetic progressions.

REFERENCES:

Allouche, J.-P. and Shallit, J. "The Ring of k-Regular Sequences." Theor. Comput. Sci. 98, 163-197, 1992.

Erdős, P. and Turán, P. "On Some Sequences of Integers." J. London Math. Soc. 11, 261-264, 1936.

Gerver, J.; Propp, J.; and Simpson, J. "Greedily Partitioning the Natural Numbers into Sets Free of Arithmetic Progressions." Proc. Amer. Math. Soc. 102, 765-772, 1988.

Guy, R. K. "Theorem of van der Waerden, Szemerédi's Theorem. Partitioning the Integers into Classes; at Least One Contains an A.P." §E10 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 204-209, 1994.

Iacobescu, F. "Smarandache Partition Type and Other Sequences." Bull. Pure Appl. Sci. 16E, 237-240, 1997.

Ibstedt, H. "A Few Smarandache Sequences." Smarandache Notions J. 8, 170-183, 1997.

Sloane, N. J. A. Sequences A003278/M0975, A033156, A033157, A033158, A033159, A033160, A033161, A033162, A033163, and A033164 in "The On-Line Encyclopedia of Integer Sequences."