Consecutive Tribonacci ratios converge to T ≈ 1.839 (red line). The sequence overshoots then oscillates in. The golden ratio φ ≈ 1.618 emerges from 2-term Fibonacci in the same way.
Both sequences grow exponentially. Their growth rate is the unique real root greater than 1 of xⁿ = xⁿ⁻¹ + … + 1. For n=2: φ. For n=3: T. The more terms you sum, the faster the growth rate, approaching 2 as n → ∞.
The Tribonacci sequence starts 0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149… Each term is the sum of the three preceding ones. The ratio of consecutive terms converges to T ≈ 1.83929. This is the Tribonacci constant, the real root of x³ = x² + x + 1.
T is the natural extension of the golden ratio φ to three terms. The golden ratio is the limit of Fibonacci ratios, where each term is the sum of the previous two. The Tribonacci constant is the limit of Tribonacci ratios, where each term sums the previous three. There is also a Tetranacci constant (four terms), a Pentanacci constant, and so on.
T is an algebraic number of degree 3. Unlike φ which has the elegant form (1+√5)/2, the Tribonacci constant requires cube roots in its closed form: T = (1 + ∛(19+3√33) + ∛(19-3√33)) / 3. It appears in the analysis of three-dimensional fractals, certain tiling problems, and the asymptotic growth of Tribonacci-related combinatorial structures.
Each row sums more previous terms. The limiting ratio increases: φ≈1.618 (2 terms), T≈1.839 (3 terms), ≈1.928 (4 terms). As n→∞, the ratio approaches 2, because with infinitely many previous terms, each new term is roughly the sum of all previous ones: halving the total each time.
The Tribonacci sequence 0, 0, 1, 1, 2, 4, 7, 13, 24, 44... has T(n) = T(n-1) + T(n-2) + T(n-3). Ratios converge to T ≈ 1.83929, the real root of x^3 = x^2 + x + 1. This is the 3-term analogue of the golden ratio: phi satisfies x^2 = x + 1 (2-term), T satisfies the analogous cubic (3-term). The n-anacci constant generalises this to n terms. The Tribonacci constant is algebraic, degree 3.