(1 + √2)ⁿ approaches integer values for large n.

eg
(1 + √2)²⁰ = 1331713.9999992...
(1 + √2)⁴⁵ = 4940866263896162.0000000000000002...

The number of 9s or 0s after the decimal point just keeps on growing.

Peculiar!

(1+sqrt(2))ⁿ
=(sqrt(2))ⁿ(1 + 1/sqrt(2))ⁿ

As n gets large, (1 + 1/sqrt(2))ⁿ becomes very close to

1 + n/sqrt(2)

so (1 + sqrt(2))ⁿ approaches

(sqrt(2))ⁿ(1 + n/sqrt(2))

Again, as n gets large, this is approximately:

n(sqrt(2))^n-1

If n is odd, then this is clearly an integer.

I'm working on the n is even case...