Residues derived from other works

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x_ = rac{ }{ }$$ we have $\lim_{n ightarrow\infty} x_n = \sqrt{ }$. the related recurrence $$p(n) = n+ , q(n) = ( n+ ), r(n)= ( n+ )$$ $$\mbox{with } x_ = , x_ = $$ yields $$\lim_{n ightarrow\infty} rac{x_n}{ ^n} = \sqrt{ }$$ and in addition $x_n$ is an integer. comment these formulas (and tons of other

= rac{lpha- }{lpha}, x_ = rac{(lpha- ) ( lpha- )}{ lpha^ }$ to arbitrary values $x_ = a, x_ =b$, assuming here that $lpha= $: $$\lim_{n ightarrow\infty} x_n = ( - \log lpha)\cdot a + ( \log lpha ) \cdot b.$$ you may try proving this formula. it was obtained empirically, i haven't proved it. and it works...

https://math.stackexchange.com/questions/3491347/simple-recurrences-converging-to-log-2-pi-e-sqrt2-and-so-on

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