# Definition:Tribonacci Sequence

## Contents

## 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

**, according to taste.**

*trib*-bo-*nar*-chiThe 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