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Let \begin{align*} c_n &= n!\left(e-\sum_{k=0}^n \frac{1}{k!}\right) \\ \\ u_n &= \bigg\lfloor{\frac{1}{c_n} \bigg\rfloor} \\ \\ v_n &= \bigg\lfloor{\frac{1}{1/c_n-\lfloor{u_n} \rfloor}} \bigg\rfloo …

How can we generate all integer solutions of the equation $$(qr+rp+pq)(x^2+y^2+z^2) = (p^2+q^2+r^2)(yz+zx+xy),$$ given that $p,q,r$ are integers? Clearly if any one of $(x,y,z), (x,z,y), (y,z,x), ( …

Define $A=(a_n)$ and $B=(b_n)$ as follows: $a_0=1$, $a_1=2$, $b_0=3$, $b_1=4$, and $$a_n=a_1b_{n-1}-a_0b_{n-2} + 2n$$ for $n \geq 2$, where $A$ and $B$ are increasing and every positive integer occurs …

Define $A=(a_n)$ and $B=(b_n)$ as follows: $a_0=1$, $a_1=2$, $b_0=3$, $b_1=4$, and $$a_n=a_0b_{n-1}+a_1b_{n-2}$$ for $n \geq 2$, where $A$ and $B$ are increasing and every positive integer occurs exac …

Suppose that $a_0 < a_1,$ $b_0 < b_1,$ and $$a_n=a_1b_{n-1}+a_0b_{n-2}+qn+r$$ for $n \geq 2$, where $a_0,a_1,b_0,b_1,q,r$ are integers such that $(a_n)$ and $(b_n)$ are increasing and ${(|a_n|)}$ and …

A power tower of a number $x$ is typified by $$ x^{x^{x^{x^{x^{x^{x^{x^{x^x}}}}}}}}.$$ Here, however, we take the liberty of referring to the set $T$ of "$\{2,3\}$-power towers"; i.e., numbers $$ …