# Tribonacci number

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A member of the Tribonacci sequence. The formula for the $n$-th number is given by A. Shannon in [a1]: $$T_n = \sum_{m=0}^{[n/2]} \sum_{r=0}^{[n/3]} \binom{ n-m-2r }{ m+r }\binom{ m+r }{ r }$$
Binet's formula for the $n$-th number is given by W. Spickerman in [a2]: $$T_n = \frac{\rho^{n+2}}{ (\rho-\sigma)(\rho-\bar\sigma) } + \frac{\sigma^{n+2}}{ (\sigma-\rho)(\sigma-\bar\sigma) } + \frac{\bar\sigma^{n+2}}{ (\bar\sigma-\rho)(\bar\sigma-\sigma) }$$ where $$\rho = \frac{1}{3}\left({ (19+3\sqrt{33})^{1/3} + (19-3\sqrt{33})^{1/3} +1 }\right)\,,$$ and $$\sigma = \frac{1}{6}\left({ 2-(19+3\sqrt{33})^{1/3} - (19-3\sqrt{33})^{1/3} }\right) + \frac{\sqrt3 i}{6}\left({ (19+3\sqrt{33})^{1/3} - (19-3\sqrt{33})^{1/3} }\right) \ .$$