[Paper Review] Generalized metallic means

Juan B. Gil, Aaron Worley|arXiv (Cornell University)|Jan 9, 2019

Advanced Mathematical Theories and Applications4 citations

TL;DR

This paper introduces generalized metallic means of arbitrary degree $k \geq 2$ by applying the invert transform to the $k$th-order Fibonacci sequence, extending the classical quadratic metallic means (e.g., golden, silver, bronze ratios). The key result is that each $m$th generalized metallic mean $\varrho_m$ is the unique positive real root of a $k$th-degree polynomial, and the ratio of consecutive terms in the resulting sequence converges to $\varrho_m$, generalizing the classical limit behavior of Fibonacci-like sequences.

ABSTRACT

The metallic means (also known as metallic ratios) may be defined as the limiting ratio of consecutive terms of sequences connected to the Fibonacci sequence via the INVERT transform. For example, the Pell sequence (invert transform of the Fibonacci sequence) gives the so-called silver mean, and the invert transform of the Pell sequence leads to the bronze mean. The limiting ratio of the Fibonacci sequence itself is known as the golden mean or ratio. We introduce a new family of kth-degree metallic means obtained through invert transforms of the generalized kth-order Fibonacci sequence. As it is the case for k=2, each generalized metallic mean is shown to be the unique positive root of a kth-degree polynomial determined by the sequence.

Motivation & Objective

• To extend the concept of metallic means—originally defined for quadratic sequences like the Fibonacci and Pell sequences—to higher-order linear recurrences of degree $k \geq 2$.
• To define a new family of $k$th-degree metallic means through the invert transform of the generalized $k$th-order Fibonacci sequence.
• To establish that each such metallic mean is the unique positive real root of a $k$th-degree polynomial derived from the recurrence relation.
• To prove that the ratio of consecutive terms in the resulting sequence converges to the corresponding metallic mean, generalizing the classical golden ratio limit.

Proposed method

• Define the generalized $k$th-order Fibonacci sequence via the recurrence $g_n = g_{n-1} + g_{n-2} + \cdots + g_{n-k}$ with initial conditions $g_0 = 0$, $g_1 = 1$, $g_2 = \dots = g_{k-1} = 1$, and generating function $G(x) = \frac{x}{1 - x - x^2 - \cdots - x^k}$.
• Apply the invert transform to generate a new sequence $g^{(m)}_n$ with generating function $G_m(x) = \frac{x}{1 - mx - x^2 - \cdots - x^k}$, corresponding to a recurrence with coefficient $m$ on the first-order term.
• Derive the characteristic polynomial $p_m(x) = x^k - m x^{k-1} - x^{k-2} - \cdots - 1$, whose unique positive real root $\varrho_m$ defines the $m$th metallic mean of degree $k$.
• Prove that $\varrho_m \in (m, m+1)$ by showing $p_m(m) < 0 < p_m(m+1)$, ensuring existence and uniqueness of the positive root.
• Establish that all roots of $p_m(x)$ are simple by analyzing the derivative of a related polynomial $q(x) = (x-1)p_m(x)$, confirming no multiple roots.
• Use the closed-form solution of the linear recurrence to show that the ratio $g^{(m)}_{n+1}/g^{(m)}_n \to \varrho_m$ as $n \to \infty$, since all other roots have magnitude strictly less than $\varrho_m$.

Experimental results

Research questions

• RQ1 How can the concept of metallic means be generalized beyond the quadratic case to sequences of higher order $k \geq 3$?
• RQ2 What is the algebraic structure of the limiting ratio for sequences generated by the invert transform of $k$th-order Fibonacci sequences?
• RQ3 Does the ratio of consecutive terms in such generalized sequences converge to a unique positive real root of a $k$th-degree polynomial?
• RQ4 Can the classical convergence result for the golden ratio be extended to higher-order sequences with colored or weighted tiles?
• RQ5 What combinatorial interpretations exist for the generalized metallic means and their associated sequences?

Key findings

• The $m$th metallic mean of degree $k$, denoted $\varrho_m$, is the unique positive real root of the polynomial $x^k - m x^{k-1} - x^{k-2} - \cdots - 1$, and lies in the interval $(m, m+1)$.
• For $k=3$, the cubic metallic means $\tau_m$ are defined as the unique positive roots of $x^3 - m x^2 - x - 1$, with $\tau_1 \approx 1.839$ (tribonacci constant), $\tau_2 \approx 2.547$, and $\tau_3 \approx 3.383$.
• The ratio of consecutive terms in the sequence $g^{(m)}_n$ converges to $\varrho_m$, as the dominant root $\varrho_m$ in the closed-form solution of the linear recurrence overwhelms the contributions from the other $k-1$ roots.
• All roots of the characteristic polynomial $p_m(x)$ are simple, ensuring the asymptotic behavior is governed solely by the dominant root $\varrho_m$, which is essential for convergence of the ratio.
• Combinatorially, $g^{(m)}_n$ counts the number of ways to tile an $(n-1) \times 1$ board using $1 \times 1$, $2 \times 1$, ..., $k \times 1$ tiles, where $1 \times 1$ tiles have $m$ available colors.
• For $k=3$ and $m=2$, the sequence $T^{(2)}_n$ begins $1, 2, 5, 13, 33, 84, 214, \dots$, illustrating the combinatorial interpretation of the generalized metallic means.