With n=0 then \(2^0+1=1+1=2⋮̸3\)(not satisfy)

With n=1 then \(2^{2.1}+1=2+1=3⋮3\)(satisfy)

With n>1 then \(2^{2n}=\left(2^2\right)^n=4^n\equiv1^n=1\)(mod 3)\(\Rightarrow2^{2n}+1\equiv2\)(mod 3)

\(\Rightarrow2^{2n}+1⋮̸3\)(not satisfy)

Answer: n=1