Rename the expression:  A*B^n + C*D^n

C is the conjugate of A

D is the conjugate of B

For the moment, ignore A and C.

If you were to expand B^n and D^n, the 'odd' terms (all integers) would be doubled and the 'even' terms (each of which includes √2) would cancel out. 

Now reintroduce A and C, but one term at a time.

2(3 - 2√2)^n + 2(3 + 2√2)^n  must be an integer as explained just above.

√2(3 - 2√2)^n - √2(3 + 2√2)^n   Now the 'odd' terms (which all include a √2 term) cancel out and all the 'even' terms (which are integers) are doubled.

For  n = {1,2,3,4,5,...}  the expression evaluates to:

oeis A077445 which is "Numbers k such that (k^2 - 8)/2 is a square."

[4, 20, 116, 676, 3940, 22964, 133844, 780100, 4546756, ...]

The same sequence is produced for n = {0, -1, -2, -3, ...}