I am doing a work on Liouville numbers. The Liouville constant $\ell=\sum_{k\geq 0}10^{-k!}$ has its approximation by rational numbers related to the fact that for $v_n=n!$, then $v_{n+1}/v_n$ tends to infinity as $n\to \infty$.

I am try to construct a more special class of Liouville numbers and for that I would like to construct an integer sequence $v_n$ such that $v_{n+1}/v_n=O(n)$ and $v_{2n+2}/v_{2n}=O(n)$ (i.e., the same order). See that for $v_n=n!$ this does not happen, since $v_{n+1}/v_n=n+1$ while $v_{2n+2}/v_{2n}=O(n^2)$.

Someone please may help me in this task?

I am doing a work on Liouville numbers. The Liouville constant $\ell=\sum_{k\geq 0}10^{-k!}$ has its approximation by rational numbers related to the fact that for $v_n=n!$, then $v_{n+1}/v_n$ tends to infinity as $n\to \infty$.

I am try to construct a more special class of Liouville numbers and for that I would like to construct an integer sequence $v_n$ such that

$c_1n<v_{n+1}/v_n<c_2n$ and $d_1n<v_{2n+2}/v_{2n}<d_2n$ (for some positive constants $c_1,c_2, d_1, d_2$). 

See that for $v_n=n!$ this does not happen, since $v_{n+1}/v_n=n+1$ while $v_{2n+2}/v_{2n}=O(n^2)$.

Someone please may help me in this task?