The 11-2-1 condition

Let's call $11^{n}+2^{n}+1=F(n)$ .For some n, let $F(n$ ) divide $F(n+1)$ .Thus, $F(n)|F(n+1)-11.F(n)\Rightarrow F(n)|-9.2^{n}-10$ and that means that $F(n)|9.2^{n} +10\Rightarrow F(n)\le9.2^{n} +10$ .So we get $11^{n}\le8.2^{n}+9$ .But we can easily prove by induction that for every natural number n>1, $11^{n}>8.2^{n}+9$ .Then we are left with only n=1 and we see that it satisfies the condition so the answer is $\boxed1$