View source for Tribonacci number

A member of the [[Tribonacci sequence]]. The formula for the $n$-th number is given by A. Shannon in [[#References|[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 [[#References|[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====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Shannon,   "Tribonacci numbers and Pascal's pyramid"  ''The Fibonacci Quart.'' , '''15''' :  3  (1977)  pp. 268–275  {{ZBL|0385.05006}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Spickerman,   "Binet's formula for the Tribonacci sequence"  ''The Fibonacci Quart.'' , '''20'''  (1981)  pp. 118–120  {{ZBL|0486.10011}}</TD></TR>
</table>

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