Sum of binomial coefficients with index divisible by 4

Question

I am trying to obtain a closed form expression for $\sum {n \choose 4k}$

I am trying to use the binomial expansion of $(1 + i)^n$ and $(1 - i)^n$. $$(1 + i)^n + (1 - i)^n = 2\left(\sum {n \choose 4k} -\sum {n \choose 4k + 2}\right)$$ Stuck at this now. I can't come up with a way to simplify the second term on RHS. Any help would be appreciated.

Answer 1

After posting this question, it struck me that

RHS = $4\sum {n \choose 4k} -2\sum {n \choose 4k} -2\sum {n \choose 4k + 2}) = 4\sum {n \choose 4k} - 2 \sum {n \choose 2k} = 4\sum {n \choose 4k} - 2 * 2^{n - 1} = 4\sum {n \choose 4k} - 2^{n} \implies \sum {n \choose 4k} = \frac{(1 + i)^n + (1 - i)^n + 2^n}{4}$

Answer 2

We have the followings :

$$(1+1)^n=\binom n0+\binom n1+\binom n2+\binom n3+\binom n4+\binom n5+\cdots$$ $$(1-1)^n=\binom n0-\binom n1+\binom n2-\binom n3+\binom n4-\binom n5+\cdots$$ $$(1+i)^n=\binom n0+\binom n1i-\binom n2-\binom n3i+\binom n4+\binom n5i-\cdots$$ $$(1-i)^n=\binom n0-\binom n1i-\binom n2+\binom n3i+\binom n4-\binom n5i-\cdots$$

Adding these gives $$(1+1)^n+(1-1)^n+(1+i)^n+(1-i)^n=4\left(\binom n0+\binom n4+\binom n8+\cdots\right)$$ I think that you can continue from here.