Dizzy integer enumeration
SOGL V0.12, 8 6 bytes
I».»⌡±

Try it Here! or try the first couple numbers (changed a bit so it'd work)
0-indexed.
Explanation:
I       increment the input
 »      floor divide by 2
  .     push the original input
   »    floor divide by 2
    ⌡   that many times
     ±    negate

Or simpler:
(input + 1) // 2 negated input // 2 times
        I     »     ±      .     »    ⌡