Finding the nth element of generalized Fibonacci sequences
The easiest way to simplify this is to come up with a method to represent the various sequences efficiently. For example, the following represents each as a tuple of starting values, and a function that takes one tuple and returns another (the next step):

SEQUENCES = {
    'fibonacci': ((0, 1), lambda a, b: (b, a+b)),
    'tribonacci': ((0, 1, 1), lambda a, b, c: (b, c, a+b+c)),
    ...
}


The function to apply these is then a trivial dictionary lookup and a loop over the number of steps:

def fibgolf(name, n):
    """Calculate the nth value of the named sequence."""
    vals, func = SEQUENCES[name]
    for _ in range(n):
        vals = func(*vals)
    return vals[0]