Inverse transform of −3.894z/(z²+0.6065)

The denominator has complex poles on the imaginary axis. So match $X(z)$ to the damped-sine pair:

$$\mathcal{Z}\left[r^k\sin k\Omega\right] = \frac{rz\sin\Omega}{z^2 - 2rz\cos\Omega + r^2}$$

Step 1: Find the poles

$$z^2 + 0.6065 = 0 \;\Rightarrow\; z = \pm j\sqrt{0.6065} = \pm j\,0.7788$$

So $r = 0.7788$ and $\Omega = \dfrac{\pi}{2}$. Note that $0.7788 = e^{-0.25}$.

Step 2: Match coefficients

With $\Omega = \pi/2$: $\cos\Omega = 0$ and $\sin\Omega = 1$. The pair becomes

$$\mathcal{Z}\left[r^k\sin\frac{k\pi}{2}\right] = \frac{rz}{z^2 + r^2} = \frac{0.7788z}{z^2 + 0.6065}$$

Rewrite $X(z)$:

$$X(z) = -\frac{3.894}{0.7788}\cdot\frac{0.7788z}{z^2 + 0.6065} = -5.0\cdot\frac{0.7788z}{z^2+0.6065}$$

since $3.894/0.7788 = 5.000$.

Step 3: Inverse transform

$$x(k) = -5\,(0.7788)^k\sin\frac{k\pi}{2} = -5\,e^{-0.25k}\sin\frac{k\pi}{2}, \quad k \ge 0$$

Step 4: Check by long division

$X(z) = \dfrac{-3.894z^{-1}}{1 + 0.6065z^{-2}} = -3.894z^{-1} + 2.3617z^{-3} - 1.4324z^{-5} + 0.8687z^{-7} - \dots$

The formula gives the same values; for example, $k = 3$: $-5(0.4724)(-1) = 2.362$.

The poles have magnitude $0.7788 < 1$, so the oscillation decays.

Answer: $x(k) = -5(0.7788)^k\sin\dfrac{k\pi}{2}$, i.e. $0, -3.894, 0, 2.362, 0, -1.432, \dots$