In addition, I've made an attempt to use $$$\vec{V_{0}}$$$ instead of $$$\vec{V_{-1}}$$$, however the result is also incorrect with my calculations:

For the particular $$$n=5$$$ it gives incorrect result:

Thus, actual result is $$$x_5=23$$$ and $$$y_5=43$$$ which is incorrect, expected result is $$$x_5 = 631$$$ and $$$y_5=5723$$$.

I would really appreciate it if you could point out my mistake.

P.S. Sorry for posting the previous reply without spoilers.

Thank you for your reply tugsuu!

I've tried to do it this way, but the result of $$$T^{n+1} \cdot \vec{V_{-1}}$$$ was incorrect:

Pretty cool! Could you explain for nubbie please)?