Dynamic Programming: Fibonacci-like recurrence relation
Recursive Solution

Code-wise, I see no issues. The one thing I'd point out is that it might be clearer to name your functions A and B instead of Nth and Aux, since the problem has to do with finding A and it isn't readily apparent what Aux means. It's a little unclear, but not a dealbreaker. 

Bottom-up method

I don't really understand why your memoized map contains an array. If it's really a "pair" type such that memoizedMap[n] is \$(A_n, B_n)\$, then I'd introduce a new type for that. Arrays can mean anything, so once you make the decision to go with an array, the code is very difficult to follow. You're introducing some boolean flags, which are probably unnecessary too. 

That said, I'm not sure that simply memoizing the recursive solution is what the problem intended. The intent seems to instead suggest that you compute the arrays iteratively. That is just the same way you could compute the Fibonacci sequence iteratively (in fact, the two sequences are very closely related, as \$A_1, B_2, A_3, B_4, ...\$ is the Fibonacci sequence): just keep the last few elements and update them as you go:

public static int getAn(int n) {
    if (n &lt;= 2) return 1;

    int a0 = 1;
    int a1 = 1;
    int b0 = 2;
    int b1 = 2;

    for (int i = 0; i &lt; n-2; ++i) {
        int nexta = b1 + a0;
        int nextb = a1 + b0;

        // updates
        a0 = a1;
        a1 = nexta;
        b0 = b1;
        b1 = nextb;
    }

    return a1;
}