Consider sequences of positive real numbers of the form x, 2000, y ..., in which every term after the first is 1 less than the product of its two immediate neighbors. For how many different values of x does the term 3000 appear somewhere in the sequence?
If I understand your question correctly, the "generating function" is as follows:
a(n) =[a(n - 1) * a(n - 2)] - 1. So, in order to get 3,000 for the 3rd term, you have the following:
x * 2,000 - 1 = 3,000
x =3,001 / 2,000
x =1.5005 - This is the only positive real number that will give 3,000 as the 3rd term. After that the sequence "blows up"!, in the sense that the 4th term will be : 2,000 x 3,000 - 1 =5,999,999......and so on.