Maclaurin series for sinx

Question

I'm learning about taylor series and I tried coming up with one (a Maclaurin) for $\sin x$. I came up with the following: $$\sin x = \sum_{n=1}^\infty (n\bmod 2) (-1)^?\frac{x^n}{n!}$$ This way, all the even $n$ terms become $0$ and excluded. However, I can't figure out which power to put on the $(-1)$ term such that the odd $n$ terms alternate between being positive and negative. I thought about using the Fibonacci sequence where it turns out that starting from $F(1)$, every third number is odd, and starting from $F(2)$ every third number is even. This would work if the pattern existed for every fourth number, rather than third. Is there any way to fill in that question mark? Thanks.

Answer 1

$?=\frac{n-1}{2}$ works.

Now, if you denote $k=\frac{n-1}{2}$, your formula becomes a nice expression in $k$, which is the usual one.

Now, if you want an expression which is integer also for even $n$, you could use $\frac{n(n-1)}{2}$ which is the same as $1+2+..+n-1$.