Somos Sequence

The Somos sequences are a set of related symmetrical Recurrence Relations which, surprisingly, always give integers. The Somos sequence of order is defined by

where is the Floor Function and for , ..., . The 2- and 3-Somos sequences consist entirely of 1s. The -Somos sequences for , 5, 6, and 7 are giving 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, ... (Sloane's A006720), 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, ... (Sloane's A006721), 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, ... (Sloane's A006722), 1, 1, 1, 1, 1, 1, 3, 5, 9, 17, 41, 137, 769, ... (Sloane's A006723). Gale (1991) gives simple proofs of the integer-only property of the 4-Somos and 5-Somos sequences. Hickerson proved 6-Somos generates only integers using computer algebra, and empirical evidence suggests 7-Somos is also integer-only.

However, the -Somos sequences for do not give integers. The values of for which first becomes nonintegral for the -Somos sequence for , 9, ... are 17, 19, 20, 22, 24, 27, 28, 30, 33, 34, 36, 39, 41, 42, 44, 46, 48, 51, 52, 55, 56, 58, 60, ... (Sloane's A030127).

References

Buchholz, R. H. and Rathbun, R. L. An Infinite Set of Heron Triangles with Two Rational Medians.'' Amer. Math. Monthly 104, 107-115, 1997.

Gale, D. Mathematical Entertainments: The Strange and Surprising Saga of the Somos Sequences.'' Math. Intel. 13, 40-42, 1991.

Sloane, N. J. A. Sequences A030127, A006720/M0857, A006721/M0735, A006722/M2457, A006723/M2456 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.