Dynamic programming with Fibonacci
A few things:

Naming:

It's neither Fibonaci, nor febonani, nor fibonanci it's fibonacci. Please get your names to reflect what you're actually talking about and not some disfigured mutation of it :(

memoized OTOH is a relatively nice name, I'd probably prefer memoizedFibonacciNumbers, but that's a thing of preference

Calculating:

quoting wikipedia:


  By definition, the first two numbers in the Fibonacci sequence are \$1\$ and \$1\$, or \$0\$ and \$1\$, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two.


You on the other hand say:


  The value of Fibonacci for \$n &lt;= 0\$ is \$0\$, and for \$n &lt;= 2\$ is \$1\$, and each subsequent number is the sum of the previous two.


Wikipedia also mentions "negative Fibonacci numbers". Your definition does not match the actual Fibonacci sequence definition :(

instead your fibonacci method should look like this, if you exactly follow the rules for only positive fibonacci numbers:

private static BigInteger fibonacci(Integer n, HashMap&lt;Integer, BigInteger&gt; memoized) {
    if (n &lt; 0) {
       throw new IllegalArgumentException("We assume the positive Fibonacci sequence only");
    }
    if (memoized.containsKey(n)) {
        return memoized.get(n);
    }
    if (n == 0) {
        memoized.put(n, BigInteger.ZERO);
        return memoized.get(n);
    }
    if (n == 1) {
        memoized.put(n, BigInteger.ONE);
        return memoized.get(n);
    }
    BigInteger fibonacci = fibonacci(n-1, memoized).add(fibonacci(n-2, memoized));
    memoized.put(n, finbonacci);
    return fibonacci;
}


Printing is slow:

I see you benchmarking your code. Please keep in mind, that writing something to the Console is quite slow. Your measured execution time is not matching the calculation time.

You should remove the System.out.println() in your fibonacci method.

Conditionals

Also you maybe have already realized, I removed the else in the fibonacci method. 

This is because if your code reaches that area, all other branches have returned early, and the else is quite useless.