Tribonacci to Infinity

Number Theory Level 3

Let $T$ be the nonnegative integer sequence defined by $\begin{aligned} T_0 &= 0 \\ T_1 &= 1 \\ T_2 &= 1 \\ T_n &= T_{n-1} + T_{n-2} +\ T_{n-3}\ \ (\text{for }n \geq 3). \end{aligned}$ Also let $Q = \lim_{n\to\infty}\frac{T_n}{T_{n-1}}.$ Find the value of $Q^3 - Q^2 - Q.$