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Somos sequence

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In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below. They were discovered by mathematician Michael Somos. From the form of their defining recurrence (which involves division), one would expect the terms of the sequence to be fractions, but nevertheless many Somos sequences have the property that all of their members are integers.

Recurrence equations

For an integer number k larger than 1, the Somos-k sequence ( a 0 , a 1 , a 2 , … ) {\displaystyle (a_{0},a_{1},a_{2},\ldots )} is defined by the equation

a n a n − k = a n − 1 a n − k + 1 + a n − 2 a n − k + 2 + ⋯ + a n − ( k − 1 ) / 2 a n − ( k + 1 ) / 2 {\displaystyle a_{n}a_{n-k}=a_{n-1}a_{n-k+1}+a_{n-2}a_{n-k+2}+\cdots +a_{n-(k-1)/2}a_{n-(k+1)/2}}

when k is odd, or by the analogous equation

a n a n − k = a n − 1 a n − k + 1 + a n − 2 a n − k + 2 + ⋯ + ( a n − k / 2 ) 2 {\displaystyle a_{n}a_{n-k}=a_{n-1}a_{n-k+1}+a_{n-2}a_{n-k+2}+\cdots +(a_{n-k/2})^{2}}

when k is even, together with the initial values

ai = 1 for i < k.

For k = 2 or 3, these recursions are very simple (there is no addition on the right-hand side) and they define the all-ones sequence (1, 1, 1, 1, 1, 1, ...). In the first nontrivial case, k = 4, the defining equation is

a n a n − 4 = a n − 1 a n − 3 + a n − 2 2 {\displaystyle a_{n}a_{n-4}=a_{n-1}a_{n-3}+a_{n-2}^{2}}

while for k = 5 the equation is

a n a n − 5 = a n − 1 a n − 4 + a n − 2 a n − 3 . {\displaystyle a_{n}a_{n-5}=a_{n-1}a_{n-4}+a_{n-2}a_{n-3}\,.}

These equations can be rearranged into the form of a recurrence relation, in which the value an on the left hand side of the recurrence is defined by a formula on the right hand side, by dividing the formula by an − k. For k = 4, this yields the recurrence

a n = a n − 1 a n − 3 + a n − 2 2 a n − 4 {\displaystyle a_{n}={\frac {a_{n-1}a_{n-3}+a_{n-2}^{2}}{a_{n-4}}}}

while for k = 5 it gives the recurrence

a n = a n − 1 a n − 4 + a n − 2 a n − 3 a n − 5 . {\displaystyle a_{n}={\frac {a_{n-1}a_{n-4}+a_{n-2}a_{n-3}}{a_{n-5}}}.}

While in the usual definition of the Somos sequences, the values of ai for i < k are all set equal to 1, it is also possible to define other sequences by using the same recurrences with different initial values.

Sequence values

The values in the Somos-4 sequence are

1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, 8209, 83313, 620297, 7869898, ... (sequence A006720 in the OEIS).

The values in the Somos-5 sequence are

1, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, 6161, 22833, 165713, ... (sequence A006721 in the OEIS).

The values in the Somos-6 sequence are

1, 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, 5047, 41783, 281527, ... (sequence A006722 in the OEIS).

The values in the Somos-7 sequence are

1, 1, 1, 1, 1, 1, 1, 3, 5, 9, 17, 41, 137, 769, 1925, 7203, 34081, ... (sequence A006723 in the OEIS).

The first 17 values in the Somos-8 sequence are

1, 1, 1, 1, 1, 1, 1, 1, 4, 7, 13, 25, 61, 187, 775, 5827, 14815 .

Integrality

The form of the recurrences describing the Somos sequences involves divisions, making it appear likely that the sequences defined by these recurrence will contain fractional values. Nevertheless, for k ≤ 7 the Somos sequences contain only integer values. Several mathematicians have studied the problem of proving and explaining this integer property of the Somos sequences; it is closely related to the combinatorics of cluster algebras.

For k ≥ 8 the analogously defined sequences eventually contain fractional values. For Somos-8 the first fractional value is the 18th term with value 420514/7.

For k < 7, changing the initial values (but using the same recurrence relation) also typically results in fractional values.

See also

References

  1. Mase, Takafumi (2013), "The Laurent phenomenon and discrete integrable systems" (PDF), The breadth and depth of nonlinear discrete integrable systems, RIMS Kôkyûroku Bessatsu, vol. B41, Res. Inst. Math. Sci. (RIMS), Kyoto, pp. 43–64, MR 3220414
  2. Malouf, Janice L. (1992), "An integer sequence from a rational recursion", Discrete Mathematics, 110 (1–3): 257–261, doi:10.1016/0012-365X(92)90714-Q.
  3. ^ Carroll, Gabriel D.; Speyer, David E. (2004), "The Cube Recurrence", Electronic Journal of Combinatorics, 11: R73, arXiv:math.CO/0403417, doi:10.37236/1826, S2CID 1446749.
  4. "A Bare-Bones Chronology of Somos Sequences", faculty.uml.edu, retrieved 2023-11-27
  5. Fomin, Sergey; Zelevinsky, Andrei (2002), "The Laurent phenomenon", Advances in Applied Mathematics, 28 (2): 119–144, arXiv:math.CO/0104241, doi:10.1006/aama.2001.0770, S2CID 119157629.
  6. Hone, Andrew N. W. (2023), "Casting light on shadow Somos sequences", Glasgow Mathematical Journal, 65 (S1): S87–S101, arXiv:2111.10905, doi:10.1017/S0017089522000167, MR 4594276
  7. Stone, Alex (18 November 2023), "The Astonishing Behavior of Recursive Sequences", Quanta Magazine

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