Let $F_n$ be the $n-th$ fibonacci-number, $$M_k:=\pmatrix{F_{k+3}&F_{k+1}\\F_{k+2}&F_k}$$ $$P_k:=\prod_{j=1}^k M_j$$ $$Q_k:=\frac{P_{k_{11}}}{P_{k_{21}}}$$
The goal is to find the value of the simple infinite continued fraction $$\alpha:=[1,2,1,1,2,1,1,1,2,1,1,1,1,2,\cdots]=1.3876267558043602953\cdots$$
With the above notations, we get $Q_{k}$, if we truncate after the partial sequence $[1,\cdots ,1,2]$ containing $k$ ones. For example $Q_3=[1,2,1,1,2,1,1,1,2]$
Therefore, we have $$\lim_{k\rightarrow\infty} Q_k=\alpha$$
I tried to calculate some products with Wolfram alpha and hoped to get a general formula for $P_k$, but then realized that the product (at least with the input I used) does not calculate the matrix product, but it multiplies the matrices entry-wise.
See here :
The correct value of $P_2$ is $$\pmatrix{18&7\\13&5}$$
The goal is to find a closed form for $\alpha$ or at least for the $P_k$, which might help to prove/disprove that $\alpha$ is transcendental.
Does anyone know a general formula for $Q_k$ or at least a method how I can correctly calculate the matrix product with Wolfram Alpha ?