Tribonacci sequence

An extension of the sequence of Fibonacci numbers having the form (with $a$, $b$, $c$ given constants):

$$t_0=a,t_1=b,t_2=c,$$

$$t_{n+3}=t_{n+2}+t_{n+1}+t_n(n\geq0).$$

The concept was introduced by the fourteen-year-old student M. Feinberg in 1963 in [a1] for the case: $a=b=1$, $c=2$. The basic properties are introduced in [a5], [a3], [a4], [a2].

The Tribonacci sequence was generalized in [a6], [a7] to the form of two sequences:

$$a_{n+3}=u_{n+2}+w_{n+1}+y_n,$$

$$b_{n+3}=v_{n+2}+x_{n+1}+z_n,$$

where $u,v,w,x,y,z\in\{a,b\}$ and each of the tuples $(u,v)$, $(w,x)$, $(y,z)$ contains the two symbols $a$ and $b$. There are eight different such schemes. An open problem (as of 2000) is the construction of an explicit formula for each of them.

See also Tribonacci number.

References

[a1]	M. Feinberg, "Fibonacci–Tribonacci" The Fibonacci Quart. , 1 : 3 (1963) pp. 71–74
[a2]	C. Valavigi, "Properties of Tribonacci numbers" The Fibonacci Quart. , 10 : 3 (1972) pp. 231–246
[a3]	A. Scott, T. Delaney, V. Hoggatt Jr., "The Tribonacci sequence" The Fibonacci Quart. , 15 : 3 (1977) pp. 193–200
[a4]	A. Shannon, "Tribonacci numbers and Pascal's pyramid" The Fibonacci Quart. , 15 : 3 (1977) pp. 268; 275
[a5]	I. Bruce, "A modified Tribonacci sequence" The Fibonacci Quart. , 22 : 3 (1984) pp. 244–246
[a6]	K. Atanassov, J. Hlebarova, S. Mihov, "Recurrent formulas of the generalized Fibonacci and Tribonacci sequences" The Fibonacci Quart. , 30 : 1 (1992) pp. 77–79
[a7]	J.-Z. Lee, J.-S. Lee, "Some properties of the generalization of the Fibonacci sequence" The Fibonacci Quart. , 25 : 2 (1987) pp. 111–117