Definition:Tribonacci Sequence
Definition
The Tribonacci sequence is a sequence $\left \langle {u_n}\right \rangle$ which is formally defined recursively as:
- $u_n = \begin{cases} 0 & : n = 0 \\
0 & : n = 1 \\ 1 & : n = 2 \\ u_{n - 1} + u_{n - 2} + u_{n - 3} & : n > 2 \end{cases}$
The Tribonacci sequence begins:
- $0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, \ldots$
This sequence is A000073 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
General Tribonacci Sequence
A general Tribonacci sequence is a sequence $\left \langle {u_n}\right \rangle$ which is formally defined recursively as:
- $u_n = \begin{cases} a & : n = 0 \\
b & : n = 1 \\ c & : n = 2 \\ u_{n - 1} + u_{n - 2} + u_{n - 3} & : n > 2 \end{cases}$
where $a, b, c \in \Z$ are constants.
Also defined as
Some sources define $u_0 = 0, u_1 = 1, u_2 = 1$, which produces the same sequence but offset by $1$.
Also see
- Results about Tribonacci sequences can be found here.
Linguistic Note
The word Tribonacci, in the context of Tribonacci constant and Tribonacci sequence, is a portmanteau word formed from tri, from the Greek word for three, and the name of the mathematician Fibonacci.
Hence it is pronounced trib-bo-nat-chi, or trib-bo-nar-chi, according to taste.
The word arises as a direct analogy with the Fibonacci numbers.
Sources
- 1998: John Sharp: Have You Seen This Number? (The Mathematical Gazette Vol. 82: pp. 203 – 214) www.jstor.org/stable/3620403