Tribonacci number

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 $$ \sigma = \frac{1}{6}\left({ 2-(19+3\sqrt{33})^{1/3} - (19-3\sqrt{33})^{1/3} }\right) + \frac{\sqrt3 i}{6}\left({ (19+3\sqrt{33})^{1/3} - (19-3\sqrt{33})^{1/3} }\right) \ . $$

References

[a1]	A. Shannon, "Tribonacci numbers and Pascal's pyramid" The Fibonacci Quart. , 15 : 3 (1977) pp. 268–275 Zbl 0385.05006
[a2]	W. Spickerman, "Binet's formula for the Tribonacci sequence" The Fibonacci Quart. , 20 (1981) pp. 118–120 Zbl 0486.10011