<s>[INST] Find the value $\binom {n}{0} + \binom{n}{4} + \binom{n}{8} + \cdots $, where $n$ is a positive integer. Given 
$$(1+x)^n= \binom {n}{0} + \binom{n}{1} x+ \binom{n}{2} x^2+ \cdots + \binom {n}{n} x^n.$$
Find the value $\binom {n}{0} + \binom{n}{4} + \binom{n}{8} + \cdots $, where $n$ is a positive integer.
I tried to use negative $x$ and even $i$ but could not eliminate  $\binom{n}{2} $ term.
 [/INST] Hint: What is $(1+1)^n+(1-1)^n+(1+i)^n+(1-i)^n$?
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