The Refined Gale-Robinson Polynomial Sequence Conjecture

                       by Michael Somos 11 Apr 1999

Given integers { k , p , q , r } such that 6 <= k , 0 < p < q < r <= k/2 .

Given variables { x , y , z } , define a sequence by

    a(1) = a(2) = ... = a(k) = 1 , and

           x * a(n-p)*a(n-k+p) + y * a(n-q)*a(n-k+q) + z * a(n-r)*a(n-k+r)
    a(n) = --------------------------------------------------------------- .
                                        a(n-k)

By definition, a(n) is a rational function in variables { x , y , z } with
non-negative integer coefficients.

Conjecture: 
      a(n) is always a polynomial if and only if p+q = r or k-r .
      It first fails to be a polynomial at a(n) when the denominator
      a(n-k) = x+y+z or x(x+y+z)+y+z , and when k+1 <= n-k <= k+1+p .
end.

References:

Richard K. Guy, Unsolved Problems in Number Theory, 2nd edition, page 215.
"E15 ... Raphael Robinson has observed ... x_nx_{n-k}=ax_{n-p}x_{n-k+p}+
bx_{n-q}x_{n-k+q}+cx_{n-r}x_{n-k+r}  ... appears to generate integers from
the starting values ... a>=0,b>=0,c>=0, p>=1,q>=1,r>=1, p+q+r=k."

David Gale, The Mathematical Intelligencer, Vol 13, No. 1, 1991, page 42.
"... Conjecture: For any p, q, r < k the recursion a_n a_{n-k} =
      xa_{n-p}a_{n-k+p} + ya_{n-q}a_{n-k+q} + za_{n-r}a_{n-k+r}       (7)
generates integers if and only if p, q, r can be chosen so that p+q+r=k.
(Robinson's evidence is only for the case x = y = z = 1. The arbitrary
x, y, z are my responsibility.)"



Further Conjecture by Michael Somos 12 Apr 1999 :

Given integers { k, p, q, r, s } such that 8 <= k , 0 < p < q < r < s <= k/2 .

Given variables { x, y, z, w } , define a sequence by

  a(1) = a(2) = ... = a(k) = 1 , and

       x*a(n-p)*a(n-k+p)+y*a(n-q)*a(n-k+q)+z*a(n-r)*a(n-k+r)+w*a(n-s)*a(n-k+s)
  a(n)=-----------------------------------------------------------------------.
                                      a(n-k)

Conjecture:
      a(n) always fails to be a polynomial at a(n) when the denominator
      a(n-k) = x+y+z+w or x^2+(y+z+w)*x+y+z+w and when k+1 <= n-k <= k+1+p .
end.

Comment:

The Somos-4, Somos-5, Somos-6, Somos-7 sequences are all special cases of
the Gale-Robinson sequence conjecture. After that, his sequences produce integers
and mine don't.