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Possible Duplicate:
Fibonacci, tribonacci and other similar sequences

Suppose my Tribonacci series is like this: \begin{equation} T(n) = T(n-1) + T(n-2) +T(n-3) \end{equation} with initial values $T(0) = 1, T(1) = 2$ and $T(2) = 3$, then how do I find $T(10000)$?

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    It's a duplicate in the sense that if you take the trouble to understand what's at that other question you'll have no trouble answering yours (and if you do have trouble, you can always try again).2012-09-08

1 Answers 1

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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} $