Tribonacci sequence

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

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

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

where and each of the tuples , , contains the two symbols and . 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