Spoiler for this was unexpected

If you simply calculate the value of (3+sqrt(5))

^{n}for small values of n, you get1 5.2360679775 2 27.416407865 3 143.55417528 4 751.65942022 5 3935.7398201998 6 20607.801240319 7 107903.848161115 8 564991.884005414 9 2958335.91138802 10 15490047.9323065 11 81106943.9482868 12 424681471.960495 13 2223661055.96982 14 11643240447.9769 15 60964798463.9824 16 319215828991.987 17 1671435780095.99It's not clear to me why the decimal portion should approach one as it's raised to higher powers, if that's indeed what happens. Similarly strange things happen with other expressions of the form (X+sqrt(Y))

^{n}, such as (1+sqrt(2))^{n}asymptotically bouncing between decimal values just below 0.999... and just above 0 that seem to converge toward zero/unity.