[OT] Conjectured Somos-like closed form of recurrences with polynomial coefficients

[Georgi Guninski <gguninski@gmail.com>]

As posted on mathoverflow [2]
>From [Our short paper][1]

For polynomial $F$ with integer coefficients, define the recurrence
$f(n)=F(n,f(n-1),f(n-2),...,f(n-d))$. We conjecture that
$f(n)$ satisfy Somos like sequence
$f(n)=\frac{G(f(n-1),...,f(n-d_1))}{H(f(n-1),...,f(n-d_2))}$
for polynomials $G$,$H$ which do not depend on $n$.

The strong conjecture is for all $F$ and the weak conjecture
is about $F$ which is linear in $f(n-i)$.

It solves special parametric cases like
$f(n)=F(n,f(n-1),f(n-2))=(f(n-1)^2)(b_1 n+b_2)+f(n-2)(b_3 n+b_4)$ for
all integers $b_i$ and $f(n)=f(n-1)(n^2+n+2)+f(n-2)(3 n+ 5 )$.
The proofs are based on finding algebraic dependency of
polynomials.

>Q1 Are the conjectures true?

---
In comments was suggested existence of algebraic dependency
not depending on $n$ using resultants of consecutive terms.

This doesn't prove the conjectures, since additional linearity
"luck" is needed.

The answers give linear algebra approach, which is more human
friendly than Groebner basis.

[1]: https://www.researchgate.net/publication/382801850_Conjectured_Somos-like_closed_form_of_recurrences_with_polynomial_coefficients
[2]: https://mathoverflow.net/q/476226/12481