S(1) = x , S(2) = S(3) = 1 , S(4) = y , S(n) = x*y*z*(S(n-1)*S(n-3)+S(n-2)^2)/S(n-4) if n > 4 .

The first observation is that this definition does produce polynomials with positive integer coefficients in the three indeterminates x , y , z. The proof is by induction. Another observation is that S(n) can be easily extended to n < 1 so that it still satisfies the non-linear equation. When S(1)=S(2)=S(3)=S(4)=1, then S(n) becomes the Somos-4 sequence of integers. For more information about these sequences, see article "SomosSequence" in MathWorld. So far, I have not found a way to extend this to other Somos sequences such as the Somos 6 sequence. Finally, this polynomial sequence is universal in the sense that it can be used to express any solution of the general Theta-4 recursion:

a(n) = (p1*a(n-1)*a(n-3)+p2*a(n-2)^2)/a(n-4) .

A brief table of the polynomials is:

S(1) = x S(2) = 1 S(3) = 1 S(4) = y S(5) = y^2*z + y*z S(6) = x*y^3*z^2 + x*y^3*z + x*y^2*z^2 S(7) = x^2*y^5*z^3 + x^2*y^5*z^2 + x^2*y^4*z^3 + x*y^5*z^3 + 2*x*y^4*z^3 + x*y^3*z^3 S(8) = x^3*y^7*z^5 + x^3*y^7*z^4 + 3*x^3*y^6*z^5 + 3*x^3*y^6*z^4 + x^3*y^6*z^3 + 3*x^3*y^5*z^5 + 2*x^3*y^5*z^4 + x^3*y^4*z^5 + x^2*y^7*z^5 + 3*x^2*y^6*z^5 + 3*x^2*y^5*z^5 + x^2*y^4*z^5

If the maximum exponent of x in all terms of S(n) is taken, the sequence is

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 e(n) 1 0 0 0 0 1 2 3 5 7 9 12 15 18 22 26 30 35 40 45 51 57 63which is given by the following simple recursion:

e(n) = 1 + e(n-1) + e(n-3) - e(n-4) .

The maximum exponent sequence of y is e(n+2), and of z is e(n+1). The sum of the maximum exponents of x, y, and z is (n-2)*(n-3)/2, not coincidently a triangular number. A shifted version of the e(n) sequence is listed as A001840 in Sloane's Online Encyclopedia of Integer Sequences (OEIS). It has a simple rational generating funcion.

The polynomials factor into powers of x, y, z, and an irreducible factor F(n) which has a total degree f(n), the first few terms of which are

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 f(n) 0 0 0 0 1 2 4 6 8 11 14 18 22 26 31 36 42 48 54 61 68 76 84

A shifted version of the f(n) sequence is listed a A011858 in the OEIS and also has a simple generating function.

Based on earlier work, around December 1992 I came up with this sequence of polynomials with unusual properties. An explanation of the way in which this sequence was discovered is a long story, but it started with a study of elliptic theta functions. There is a lot more to the story which I will write up if people are interested in it.

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Michael Somos <ms639@georgetown.edu>

Michael Somos