There is no clever method. Associated characteristic equation $ x^3-x^2-x-1=0 $ got a real solutions $\varphi$ and two conjuguate complex solutions $\psi,\overline{\psi}$. Using an online cubic equation solver yields : $ \varphi = 1.8392867552141612 $ $ \psi = -0.41964337760708065 + 0.6062907292071992 i$ If you are looking for rational expression then take a look at this.
The Tribonacci general term is thus $ T(n) = a \varphi^n + b \psi^n + c\left(\overline{\psi}\right)^n $
Your initial conditions gives you the values for $a,b,c$. Then $ T(10000) = a \varphi^{10000} + b \psi^{10000} + c\left(\overline{\psi}\right)^{100000} $