Somos Sequence

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

$$a_n={\sum_{j=1}^{\left\lfloor{k/2}\right\rfloor } a_{n-j} a_{n-(k-j)}\over a_{n-k}},$$

where $\left\lfloor{x}\right\rfloor$ is the Floor Function and $a_j=1$ for $j=0$, ..., $k-1$. The 2- and 3-Somos sequences consist entirely of 1s. The $k$-Somos sequences for $k=4$, 5, 6, and 7 are

$a_n={a_{n-1}a_{n-3}+{a_{n-2}}^2\over a_{n-4}}$
$a_n={a_{n-1}a_{n-4}+a_{n-2}a_{n-3}\over a_{n-5}}$
$a_n={1\over a_{n-6}}[a_{n-1}a_{n-5}+a_{n-2}a_{n-4}+{a_{n-3}}^2]$
$a_n={1\over a_{n-7}}[a_{n-1}a_{n-6}+a_{n-2}a_{n-5}+a_{n-3}a_{n-4}],$

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 $k$-Somos sequences for $k\geq 8$ do not give integers. The values of $n$ for which $a_n$ first becomes nonintegral for the $k$-Somos sequence for $k=8$, 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).

See also Göbel's Sequence, Heronian Triangle

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.