Definition:Tribonacci Sequence

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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