Consecutive Tribonacci ratios converge to T ~1.839 (red line). The sequence overshoots and oscillates in. The golden ratio phi ~1.618 emerges the same way from Fibonacci.
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.
Table comparing Fibonacci Tribonacci and Tetranacci sequences and their limiting ratios
| Sequence | Rule | Terms | Limit |
|---|---|---|---|
| Fibonacci | sum of 2 | 1,1,2,3,5,8,13,21... | φ≈1.618 |
| Tribonacci | sum of 3 | 1,1,2,4,7,13,24... | T≈1.839 |
| Tetranacci | sum of 4 | 1,1,2,4,8,15,29... | ≈1.928 |
| Pentanacci | sum of 5 | 1,1,2,4,8,16,31... | ≈1.966 |
| n-nacci | sum of n | ... | → 2 |
| As you sum more terms, the growth rate approaches 2 (doubling each step) |
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.