having to change the printout length because the template assumes string conversion is just cumbersome, the template for python isn't that great

Here is my Python solution!

def fibonacciModified(t1, t2, n):
    sys.set_int_max_str_digits(99999999)
    first = t1
    second = t2
    for i in range(n - 2):
        current = first + second ** 2
        first = second
        second = current
    return current

Java:

public static BigInteger fibonacciModified(int t1, int t2, int n) {
    BigInteger n2 = BigInteger.valueOf(t1), n1 = BigInteger.valueOf(t2), fib = BigInteger.ZERO;
    if (n == 1) {
        return n2;
    }
    if (n == 2) {
        return n1;
    }
    for (int i=3; i <=n; ++i) {
        fib = n2.add(n1.multiply(n1));
        n2 = n1;
        n1 = fib;
    }
    return fib;
}

For the solution in java, the method definition should be modified to BigInteger, else it won't work for big numbers.