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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 :

https://www.wolframalpha.com/input/?i=product+j%3D1,2+%5B%5Bfib(j%2B3),fib(j%2B1)%5D,%5Bfib(j%2B2),fib(j)%5D%5D

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 ?

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    This query could help https://www.wolframalpha.com/input/?i=%7B%7Bfib(j%2B3),fib(j%2B1)%7D+,%7Bfib(j%2B2),fib(j)%7D%7D.%7B%7Bfib(j%2B4),fib(j%2B2)%7D+,%7Bfib(j%2B3),fib(j%2B1)%7D%7D+for+j%3D1 – 2017-02-04
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    @kingw But to input a product with $20$ factors this way is quite cumbersome. – 2017-02-04
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    I can calculate the products with PARI/GP, but Wolfram might detect some relations. – 2017-02-04
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    Oh well maybe this could help, though I'm not sure http://mathematica.stackexchange.com/questions/15555/i-need-to-multiply-a-series-of-matrices – 2017-02-04

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