sequence[largestIndex] = random(0, 3);
You generate a random number from the set 0, 1, and 2. And you need an int array to hold those values? Are you sure you don't need a long array, or a long long array?
for (int index = 0; index < largestIndex; index++)
{
if (Serial.available() > 0) //check to see if something came in the serial port
{
Don't you think you should wait for user input? Perhaps, do nothing while(Serial.available() == 0).
Doesn't the switch statement seem silly? In each case, you simply add the entered letter to the String.