Some more facts about Tribonacci numbers, and higher sequences:
1. The ratio between 2 successive Fibonacci numbers approaches the golden mean or golden ratio, (sqrt(5)+1)/2, which is a root of x*x - x - 1 = 0. The equivalent ratio between 2 successive Tribonacci numbers is 1.839286755, which is a root of x*x*x - x*x - x - 1 = 0.
2. As with the Fibonacci numbers, you can vary the starting values to give different sequences. The standard Tribonacci numbers use starting values 0,0,1 to give a sequence beginning 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927...
3. However, if you start 1,1,1 the resulting sequence is all odd numbers: 1, 1, 1, 3, 5, 9, 17, 31... The proof of this result is trival when done by induction, based on the fact that the sum of 3 odd numbers is itself odd.
4. The analogous sequence summing the last 4 values is called the Tetrabonacci numbers. It begins 1, 1, 2, 4, 8, 15, 29, 108, and the ratio of successive numbers converges to 1.92756
5. Another way of constructing a sequence is to add all the previous values. In this case, you get 1, 1, 2, 4, 8, 16, 32... This is simply the sum of powers of 2, and corresponds graphically to the structure of a binary tree. The ratio between successive terms in this sequence converges very rapidly to 2
6. Although I have no proof, I assume that in all series generated by summing the previous n values, the ratio between 2 successive terms will converge to a particular ratio.
7. From the last two points, it is obvious that 2 is the upper limit for the converging ratio between terms in any general n-bonacci sequence, and the higher n is (the more successive terms you add), the closer the ratio will get to the limit of 2.
8. Tribonacci numbers can also be used to win at the Four Numbers Game.
Further reading
Brian Hayes, "The Vibonacci Numbers". American Scientist, http://www.sigmaxi.org/amsci/issues/comsci99/compsci1999-07.html, 2001/10/30.
Anonymous, "Odd Golden Section". http://www.kees.cc/gldsec.html, 2001/10/30.
Anonymous, "The Four Numbers Game". http://www.geocities.com/SiliconValley/Park/5376/num.html, 2001/10/30.