I'm trying figure out a formula that would work with this sequence defined by a0=1, a1=2, and ak = 2ak-1 + 3ak-2, for all integers k is equal or greater than 2.

I found the values for the rest of the a subscripts

a2 = 7

a3 = 20

a4 = 61

a5 = 182

Seems like it's the values are 3 times more than previous value plus 1 or minus 1

As for de Morgan's law for unions.

If A and B are subsets of the universal set U, then (AUB)^c = (A^c)⋂(B^c).

Would i prove this law by showing a universal set then select random values from the universal set to set A and set B? After I would show AUB, A^c, B^c, etc all the way to (AUB)^c = (A^c)⋂(B^c)?

or would i need to prove this by some form of induction?....