Tribonacci to Infinity

Number Theory Level 3

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