Recurrence relationship $F(n+2)-2F(n+1)-8F(n)=0$

Question

$$F(n+2)-2F(n+1)-8F(n)=0$$ Our solution is trigonometric algebraic $$ 2^n(2^n-\cos(\pi n)) $$ My question is as fallow, can we find solution to this recurrence relation that has solution $$ F^\text{(m)}(n)=F(n) $$ where $m$ is mth derivative mainly ($m=0,1,2,3$) and this two are not equal every where conditionally most specifically where $n=0,1,2,3,\dots$
What is the best way to go about this?.