As requested, I elaborate on my comment.

First of all, let me make a change of variables $a_i=U_i-1$. The relations then become $a_i+1=a_{i-1}a_{i+1}$.

For $n=2,3,4$ I used the Magma online calculator. The commands

```
F<x,y> := FreeAlgebra(RationalField(),2);
B := [x^2-y-1,y^2-x-1];
GroebnerBasis(B);
```

give the result

```
[
x^2 - y - 1,
x*y - y*x,
y^2 - x - 1
]
```

so the algebra has a basis $1,x,y,yx$, so is four-dimensional.

The commands

```
F<x,y,z> := FreeAlgebra(RationalField(),3);
B := [x*z-y-1,y*x-z-1,z*y-x-1];
GroebnerBasis(B);
```

give the result

```
[
x*y*z - z*x*y + y*z - z*x - x + y,
y*z*x - z*x*y - x*y + y*z - x + z,
y*z^2 + y*z - z*x - x - z - 1,
z^2*x - y*z + z*x - y - z - 1,
z^3 - x*y + z^2 - x - y - 1,
x^2 - z^2 + x - z,
x*z - y - 1,
y*x - z - 1,
y^2 - z^2 + y - z,
z*y - x - 1
]
```

so the algebra has a basis $1,x,y,z,xy,yz,zx,z^2,zxy$, so is nine-dimensional.

A similar computation for $n=4$ gives a Gröbner basis which is a bit too long to format properly, and a basis for the algebra $1,x,y,z,t,xy,yz,zt,tx,tz,x^2,y^2,z^2,t^2,xyz,y^2z,yzt,z^2t,ztx,tx^2,txy,t^2x,t^3,txyz,t^2x^2$, so the algebra is 25-dimensional.

For $n=5$, the calculator spits more and more elements as the degree grows, so it might even be that the Gröbner basis is infinite. However, if we consider the abelianisation of this algebra, the command

```
F := PolynomialRing(RationalField(),5);
B := [x*z - y-1, y*t-z-1, z*u-t-1, t*x-u-1, u*y-x-1];
GroebnerBasis(B);
```

produces the result

```
[
x - y*u + 1,
y*t - z - 1,
z*u - t - 1
]
```

which defines an infinite-dimensional algebra (since the associated monomial algebra is defined by the relations $x=0$, $yt=0$, $zu=0$), so the original algebra must for sure be infinite-dimensional.

[Alternatively, we can note that for commuting elements $a_i$, the equation $a_{i-1}a_{i+1}=a_i+1$ is the celebrated "pentagon recurrence" related to cluster algebras of type $A_2$, and we have $a_{n+5}=a_n$ for all choices of $a_0,a_1$ for which the sequence is uniquely defined, so the corresponding abelianisation corresponds to something 2-dimensional geometrically, and the algebra is infinite-dimensional.]

Overall, it is not quite clear if we should hope that these algebras obey a nice pattern or are easily recognisable, but they do look interesting.