Solve Pascals Triangle Even Number Sequence w/Fibonacci & Combinations

[Noir]

Homework Statement

In Pascals triange, when you colour in the even numbers you get what is known as the Sierpinski triangle. My task is find two general formula's that link the number in a specific sequence with fibonacci and combinations. The sequence is the amount of coloured in even numbers in the largest cluster in the triangle. The sequence, is "1,6,28,120,456,2016..."

The Attempt at a Solution

I had a hunch about the combination one, where because all evens are divisible by 2, that was apart of it. The formula i got was
Area (or number of evens) = 2^n C 2
(where n is the number in the sequence)
6 is the second number is the sequence, so;
2^2 C 2 = 4 C 2 = 6.
So i have this bit right, but I'm totally stumpted on the fibonacci formula. Can anyone help me or point me in the right direction? I hope I've given enough information for you all, thank you :)

 

[foxjwill]

what do you mean by the largest cluster?

edit: scratch that. I figured that part out.

 

[Noir]

Thanks for your reply. I did some work and got a formula "F3n - n!", but it only worked for the first 4 terms in the sequence...