Tribonaci et al

The tribonacci numbers are like the Fibonacci numbers, but instead of starting with two predetermined terms, the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms. The first few tribonacci numbers are:

0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, … (sequence A000073 in the OEIS)

The tribonacci constant ${\displaystyle {\tfrac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}}$ is the ratio toward which adjacent tribonacci numbers tend. It is a root of the polynomial x³ − x² − x − 1, approximately 1.839286755214161 (sequence A058265 in the OEIS), and also satisfies the equation x + x⁻³ = 2. It is important in the study of the snub cube.
The tribonacci numbers are also given by^[4]

${\displaystyle T(n)=\left\lfloor 3\,b{\frac {\left({\frac {1}{3}}\left(a_{+}+a_{-}+1\right)\right)^{n}}{b^{2}-2b+4}}\right\rceil }$

where ${\displaystyle \lfloor \cdot \rceil }$ denote the nearest integer function and

${\displaystyle a_{\pm }=\left(19\pm 3{\sqrt {33}}\right)^{1/3}}$

${\displaystyle b=\left(586+102{\sqrt {33}}\right)^{1/3}}$.

http://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers