TITLE: Denominators of certain Laurent polynomials
QUESTION [10 upvotes]: Consider the following somos-like sequence
$$x_n=\frac{x_{n-1}^2+x_{n-2}^2}{x_{n-3}}.$$
It's known that $x_n$ is a Laurent polynomial in $x_0, x_1$ and $x_2$. I got interested in the denominators of the sequence $x_n$. Some initial observations indicate particular structures regarding the exponents of the denominators, so I am tempted to ask:

Question: Is this true?
  $$x_n=P_n(x_0,x_1,x_2)x_0^{1-F_{n-3}}x_1^{1-F_{n-4}}x_2^{1-F_{n-5}},$$
  where $P_n$ is some polynomial and $F_n$ is the Fibonacci sequence.

REPLY [4 votes]: Suppose that
 $$\, x_n = c\,x_{n-1}x_{n-2} - x_{n-3} \tag{1}$$ for all
 $\, n \in \mathbb{Z}\,$ where $\, x_0, x_1, x_2 \,$ are variables and
 $\, c := (x_0^2 + x_1^2 + x_2^2) / (x_0 x_1 x_2). \,$ Then
 $$ x_n x_{n-3} = x_{n-1}^2 + x_{n-2}^2 \tag{2}$$ for all
 $\, n \in \mathbb{Z}.\,$
Define $\, P_n := x_n\, x_0^{e_n} x_1^{e_{n-1}} x_2^{e_{n-2}} \,$
 for all $\, n \in \mathbb{Z} \,$ where $\, e_n := -1 + |F_n|. \,$
Notice that $\, x_{2-n}, P_{2-n} \,$ are the same as $\, x_n, P_n \,$
except that  $\, x_0 \,$ and $\, x_2 \,$ are swapped. Now we get
 $$ P_n = (x_0^2 + x_1^2 + x_2^2)\, P_{n-1}P_{n-2} -
 \big(x_0^{2F_{n-2}} x_1^{2F_{n-3}} x_2^{2F_{n-4}}\big)\, P_{n-3} \tag{3}$$
for all $\, n>3 \,$ from equation $(1)$. Now
 $\, P_0 = P_1 = P_2 = 1, P_3 = x_1^2 + x_2^2 \,$ and the exponents
in equation $(3)$ are non-negative, thus $\, P_n \,$ is a polynomial
in $\, x_0,x_1,x_2 \,$ for all $\, n \in \mathbb{Z} \,$ by induction.