Using the Binomial Theorem, $4^n = (1+3)^n = 1 + 3n + \binom{n}{2}3^2 + \binom{n}{3}3^3 + \dots + 3^n$. Substituting this into the expression for set $A$, we get $4^n - 3n - 1 = 9[\binom{n}{2} + 3\binom{n}{3} + \dots + 3^{n-2}]$. Since the expression in the brackets is an integer for $n \ge 2$, every element of $A$ is a multiple of 9. For $n=1, A=0$; $n=2, A=9$; $n=3, A=54$. Set $B$ consists of all non-negative multiples of 9: $B = \{0, 9, 18, 27, 36, 45, 54, \dots\}$. Every element in $A$ exists in $B$, but elements like 18 and 27 are in $B$ but not in $A$. Thus, $A$ is a proper subset of $B$ ($A \subset B$). Option 2 is false as $B$ is larger. Option 3 is false because $A \neq B$. Option 4 is false because they share common elements like 0 and 9.