Algebraic numbers among other numbers

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The golden ratio (see "The first irrational numbers"), whose value is $\frac{1+\sqrt{5}}{2}\backslash frac\{1+\backslash sqrt\{5\}\}\{2\}$ is related to the Fibonacci numbers (see "To belong, or not..."). It is the limit, as n tends to infinity, of the ratio $\frac{F_{n+1}}{F_n}\backslash frac\{F\_\{n+1\}\}\{F\_n\}$. To see why, note that this ratio has a positive limit satisfying the equation x² ‒ x ‒ 1 = 0: it really is the golden ratio!

The Fibonacci numbers can be generalized: the Tribonacci numbers are defined in the same way, by a third-order recurrence relation:

*T_n = T_n*_‒1 + *T_n*_‒2 + *T_n*_‒3 for every n ≥ 3, with T₀ = T₁ = 0 and T₂ = 1.

The first terms are therefore 0, 0, 1, 1, 2, 4, 7, 13, 24 and 44. The name chosen for these numbers reveals a certain mathematical sense of humor! So what is the limit, as n tends to infinity, of the ratio $\frac{T_{n+1}}{T_n}\backslash frac\{T\_\{n+1\}\}\{T\_n\}$? It can be shown to be the only positive root of the equation x³ ‒ x² ‒ x ‒ 1 = 0; its approximate value is 1.839286755, and its exact expression is $\frac{1+\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}}{3}\backslash frac\{1+\backslash sqrt[3]\{19+3\backslash sqrt\{33\}\}+\backslash sqrt[3]\{19-3\backslash sqrt\{33\}\}\}\{3\}$. It is sometimes called the silver ratio (one wonders why...)! We shall call it the Tribonacci constant here, since there is another silver ratio, a member of the family of metallic ratios (see Tangente 203, 2022), defined as the positive root of the equation x² ‒ 2x ‒ 1 = 0 and equal to $1+\sqrt{2}1+\backslash sqrt\{2\}$.