Tribonacci Sequence Calculator
The Tribonacci sequence extends the Fibonacci idea to three preceding terms: each term equals the sum of the three terms immediately before it. The sequence begins T(0) = 0, T(1) = 0, T(2) = 1, and then T(n) = T(n-1) + T(n-2) + T(n-3). Enter any index n from 0 to 40 and this calculator returns T(n), displays all terms from T(0) to T(n) in a table, and shows the ratio T(n)/T(n-1) converging toward the Tribonacci constant of approximately 1.8392867552. The Tribonacci constant is the positive real root of x^3 - x^2 - x - 1 = 0, analogous to how the golden ratio is the positive root of x^2 - x - 1 = 0. Tribonacci numbers count the number of ways to climb n steps if you can take 1, 2, or 3 steps at a time (with an offset), and they appear in combinatorics, algorithm analysis, and the geometry of certain Archimedean solids. Values grow roughly as (1.839)^n, reaching tens of billions by n = 40.
Tribonacci sequence table (first n+1 terms)
| n | T(n) | Ratio T(n)/T(n-1) |
|---|
Tribonacci formula
T(0) = 0, T(1) = 0, T(2) = 1
T(n) = T(n-1) + T(n-2) + T(n-3) for n ≥ 3
Tribonacci constant t: real root of x^3 - x^2 - x - 1 = 0
t = 1.8392867552141612...
Tribonacci: frequently asked questions
What is the Tribonacci sequence?
The Tribonacci sequence is a generalisation of the Fibonacci sequence where each term is the sum of the three preceding terms instead of two. It starts T(0)=0, T(1)=0, T(2)=1, and then T(n) = T(n-1) + T(n-2) + T(n-3). The first several terms are: 0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81.
What is the Tribonacci constant?
The Tribonacci constant (approximately 1.8392867552) is the limit of the ratio T(n)/T(n-1) as n increases. It is the real root of the polynomial x^3 - x^2 - x - 1 = 0. The exact value is (1 + (19+3*sqrt(33))^(1/3) + (19-3*sqrt(33))^(1/3)) / 3.
How is Tribonacci different from Fibonacci?
Fibonacci adds the last two terms; Tribonacci adds the last three. This means the Tribonacci sequence grows faster than Fibonacci and converges to a different ratio (approximately 1.839 versus the golden ratio approximately 1.618). There are also Tetranacci (four terms), Pentanacci (five terms), and so on.
How large do Tribonacci numbers get?
Tribonacci numbers grow roughly as (1.839)^n. T(40) is already in the billions. By T(50) the value exceeds 10^13. This calculator handles n up to 40 using standard JavaScript numbers; beyond that, floating-point rounding errors would accumulate.
Are Tribonacci numbers used in practice?
Tribonacci numbers appear in combinatorics (counting paths of length n that can take steps of 1, 2, or 3), the analysis of certain recursive algorithms, and some areas of phyllotaxis. The Tribonacci constant appears in the formula for the side-length ratio of the snub cube.
Sources
- Tribonacci sequence and constant: Wolfram MathWorld, Tribonacci Number.
- Recurrence sequences in combinatorics: OEIS A000073 (Tribonacci numbers).
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.