Tribonacci Number — Definition, Formula & Examples

A Tribonacci number is a term in a sequence where each value equals the sum of the three terms before it, starting from 0, 0, 1. It extends the idea behind Fibonacci numbers by looking back three positions instead of two.

The Tribonacci sequence $\{T_n\}$ is defined by the recurrence relation $T_n = T_{n-1} + T_{n-2} + T_{n-3}$ for $n \geq 3$ , with initial conditions $T_0 = 0$ , $T_1 = 0$ , and $T_2 = 1$ .

Key Formula

T_n = T_{n-1} + T_{n-2} + T_{n-3}, \quad n \geq 3

Where:

$T_n$ = The nth Tribonacci number
$T_0 = 0,\; T_1 = 0,\; T_2 = 1$ = Initial conditions that start the sequence

How It Works

To find the next Tribonacci number, add the three most recent terms together. The sequence begins 0, 0, 1, and every term after that is computed from the three before it. Because each term depends on three predecessors rather than two, Tribonacci numbers grow faster than Fibonacci numbers. The ratio of consecutive terms converges to the Tribonacci constant, approximately 1.8393.

Worked Example

Problem: Find the first 10 Tribonacci numbers (T₀ through T₉).

Initial values: Start with the given seeds.

T_0 = 0,\; T_1 = 0,\; T_2 = 1

Compute T₃ through T₅: Add the three preceding terms for each new entry.

T_3 = 1+0+0 = 1,\quad T_4 = 1+1+0 = 2,\quad T_5 = 2+1+1 = 4

Compute T₆ through T₉: Continue the same process.

T_6 = 4+2+1 = 7,\quad T_7 = 7+4+2 = 13,\quad T_8 = 13+7+4 = 24,\quad T_9 = 24+13+7 = 44

Answer: The first 10 Tribonacci numbers are 0, 0, 1, 1, 2, 4, 7, 13, 24, 44.

Why It Matters

Tribonacci numbers appear in combinatorics problems that count arrangements with three choices at each step. They also show up in computer science algorithm analysis and in discrete math courses when studying higher-order linear recurrences.

Common Mistakes

Mistake: Using only two preceding terms instead of three (computing Fibonacci instead of Tribonacci).

Correction: Always sum the three most recent terms. For example, T₆ = T₅ + T₄ + T₃ = 4 + 2 + 1 = 7, not T₅ + T₄ = 6.