Question 3
For a positive integer $k$, define $A_n$ for $n=1,2,\ldots$ by
$$A_{n+1}=\frac{nA_n+2(n+1)^{2k}}{n+2},\qquad A_1=1.$$
Prove that $A_n$ is an integer for all $n\ge1$, and that $A_n$ is odd if and only if $n\equiv1$ or $2\pmod4$.
$$A_{n+1}=\frac{nA_n+2(n+1)^{2k}}{n+2},\qquad A_1=1.$$
Prove that $A_n$ is an integer for all $n\ge1$, and that $A_n$ is odd if and only if $n\equiv1$ or $2\pmod4$.