A NEW GENERALIZATION OF THE GOLDEN RATIO

Vedran Krčadinac

Department of Mathematics, University of Zagreb P.P. 335, HR-10002 Zagreb, Croatia

                        E-mail: krcko@math.hr

April 25, 2005

ABSTRACT

We propose a generalization of the golden section based on division in mean and extreme ratio. The associated integer sequences have many interesting properties.

1 GENERALIZED GOLDEN RATIOS

There have been many generalizations of the number known as golden ratio or golden section, ϕ = \frac{1+\sqrt{5}}{2}. Examples are G.A. Moore’s golden numbers [10] and S. Bradley’s nearly golden sections [5] (see also [7] and [9]). A generalization that has been considered by several authors are the positive roots of x^{k+1} − x^{k} − 1 = 0; see [12] and [14]. In this paper, a similar generalization is proposed. It is based on the original definition of ϕ, division of a line segment in mean and extreme ratio.

Let G be a point dividing the segment \overline{AB} in parts of length a = |AG| and b = |GB|; suppose a > b. The division is mean and extreme if the ratio of the larger to the smaller part equals the ratio of the whole segment to the larger part:

\frac{a}{b} = \frac{a + b}{a}.

Given a positive integer k, we consider divisions satisfying

(\frac{a}{b})^{k} = \frac{a + b}{a}.

For k > 1, we have not one but two ratios: φ_{k} = \frac{a}{b} and ϕ_{k} = \frac{a+b}{a} = 1 + \frac{1}{φ_{k}}. These numbers will be called the k-th lower and upper golden ratio,

\begin{matrix}k & φ_{k} & ϕ_{k} \\ 1 & 1.6180339887 & 1.6180339887 \\ 2 & 1.3247179572 & 1.7548776662 \\ 3 & 1.2207440846 & 1.8191725134 \\ 4 & 1.1673039783 & 1.8566748839 \\ 5 & 1.1347241384 & 1.8812714616 \end{matrix}

Table 1: Lower and upper golden ratios.

respectively. Obviously, (φ_{k})^{k} = ϕ_{k}. It is also evident that φ_{k} is a root of the polynomial p_{k}(x) = x^{k+1} − x − 1 and ϕ_{k} is a root of the polynomial P_{k}(x) = x(x − 1)^{k} − 1.

Proposition 1.1 For every positive integer k, the polynomials p_{k}(x) and P_{k}(x) have a unique positive root. If k is even this is the only real root, and if k is odd the polynomials have another negative root.

Proof. The equation p_{k}(x) = 0 can be rewritten as x^{k} − 1 = \frac{1}{x}. Thus, real roots correspond to intersections of the hyperbola y = \frac{1}{x} and the graph of the power function translated one unit downwards, y = x^{k} − 1. Similarly, real roots of P_{k} correspond to intersections of the hyperbola and the graph of the power function translated one unit to the right, y = (x − 1)^{k}. The claims follow from elementary properties of the functions involved. □

Therefore, φ_{k} is the unique positive root of p_{k} and ϕ_{k} is the unique positive root of P_{k}. The only instance when φ_{k} and ϕ_{k} coincide is k = 1, when both are equal to the ordinary golden ratio ϕ. The second lower golden ratio φ_{2} has been called plastic number by the Benedictine monk and architect Dom Hans van der Laan [1]. This is the smallest Pisot-Vijayaraghavan number (see [4]). Its square, ϕ_{2}, is also a cubic Pisot-Vijayaraghavan number. In Table 1, we list decimal approximations to the first five lower and upper golden ratios. As k grows, the lower golden ratios tend to 1 and the upper golden ratios tend to 2.

Proposition 1.2 lim_{k→∞} φ_{k} = 1, lim_{k→∞} ϕ_{k} = 2.

Proof. By direct computation, p_{k} is strictly increasing on [1, ^{k+1} \sqrt{3} ], attains a negative value at x = 1 and a positive value at x = ^{k+1} \sqrt{3}. Hence, p_{k} has a unique zero in this interval, i.e. φ_{k} ∈ (1, ^{k+1} \sqrt{3} ). The proposition follows from lim_{k→∞} ^{k+1} \sqrt{3} = 1 and ϕ_{k} = 1 + \frac{1}{φ_{k}}. □

2 ASSOCIATED INTEGER SEQUENCES

The connection between the golden ratio and Fibonacci numbers is well known. We can define integer sequences associated with the generalized golden ratios in a similar manner. The k-th lower Fibonacci sequence f_{n}^{(k)} is defined by f_{1}^{(k)} = f_{2}^{(k)} = … = f_{k+1}^{(k)} = 1 and the linear recurrence with characteristic polynomial p_{k}:

f_{n}^{(k)} = f_{n−k}^{(k)} + f_{n−k−1}^{(k)}.

The k-th upper Fibonacci sequence F_{n}^{(k)} satisfies the same initial conditions and the linear recurrence with characteristic polynomial P_{k}. By the binomial theorem, we get

F_{n}^{(k)} = ∑_{i=1}^{k} \binom{k}{i}(−1)^{i+1}F_{n−i}^{(k)} + F_{n−k−1}^{(k)}.

Of course, both f_{n}^{(1)} and F_{n}^{(1)} are just the Fibonacci numbers. The second lower Fibonacci sequence has been called the Padovan sequence in [13]:

(f_{n}^{(2)}) = (1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, …).

This is sequence number A000931 in N. Sloane’s Encyclopedia of Integer Sequences [11]. Another interesting sequence satisfying the same recurrence with different initial conditions is the Perrin sequence (Sloane’s A001608), giving a necessary condition for primality [2]. The second upper Fibonacci sequence is Sloane’s A005251:

(F_{n}^{(2)}) = (1, 1, 1, 2, 4, 7, 12, 21, 37, 65, 114, 200, 351, 616, 1081, …).

Among other combinatorial interpretations, F_{n}^{(2)} is the number of compositions of n without 2’s [6] and the number of binary strings of length n − 3 without isolated ones [3]. Notice that F_{n+1}^{(2)} = f_{2n−1}^{(2)}.

The third lower Fibonacci sequence is listed in [11] as A079398:

(f_{n}^{(3)}) = (1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 7, 8, 9, 12, 15, 17, 21, 27, 32, …).

Upper Fibonacci sequences are currently listed up to k = 5. Here are the first few values of F_{n}^{(3)}, Sloane’s A003522:

(F_{n}^{(3)}) = (1, 1, 1, 1, 2, 5, 11, 21, 37, 64, 113, 205, 377, 693, 1266, …).

De Villiers [14] considered sequences defined by the recurrence L_{n}^{(k)} = L_{n−1}^{(k)} + L_{n−k−1}^{(k)}. When equipped with Fibonacci-like initial conditions, L_{1}^{(k)} = … = L_{k+1}^{(k)} = 1, these are the Lamé sequences of higher order (according to [11]). De Villiers gave a partial proof that ratios of consecutive members tend to the positive root of x^{k+1}−x^{k}−1 = 0, generalizing a famous property of the Fibonacci numbers. The proof was later completed by S. Falcon [8]. Not surprisingly, ratios of consecutive members of the lower and upper Fibonacci sequences tend to the corresponding golden ratios.

Theorem 2.1 lim_{n→∞} \frac{f_{n+1}^{(k)}}{f_{n}^{(k)}} = φ_{k}, lim_{n→∞} \frac{F_{n+1}^{(k)}}{F_{n}^{(k)}} = ϕ_{k}.

Proof. The polynomials p_{k}, P_{k} and their derivatives are relatively prime. Therefore, p_{k} and P_{k} have k +1 distinct complex roots each and formulae for the corresponding integer sequences are of the form a_{n} = C_{0}z_{0}^{n} + … + C_{k}z_{k}^{n}. Here, z_{0}, …, z_{k} are the roots of p_{k} or P_{k} and C_{0}, …, C_{k} are constants. The quotient of two consecutive sequence members can be expressed as

\frac{a_{n+1}}{a_{n}} = \frac{C_{0}z_{0}^{n+1} + … + C_{k}z_{k}^{n+1}}{C_{0}z_{0}^{n} + … + C_{k}z_{k}^{n}} = \frac{C_{0}z_{0} + C_{1}z_{1} (\frac{z_{1}}{z_{0}} )^{n} + … + C_{k}z_{k} (\frac{z_{k}}{z_{0}} )^{n}}{C_{0} + C_{1} (\frac{z_{1}}{z_{0}} )^{n} + … + C_{k} (\frac{z_{k}}{z_{0}} )^{n}} .

Suppose |z_{0}| > |z_{i}| for i = 1, …, k. Then, (\frac{z_{i}}{z_{0}} )^{n} → 0 as n → ∞ and \frac{a_{n+1}}{a_{n}} → z_{0}, provided C_{0} ≠ 0. Thus, it remains to be shown that the coefficients C_{0}, …, C_{k} are not zero and φ_{k}, ϕ_{k} are greater than the absolute values of the remaining roots of p_{k} and P_{k}.

The coefficients C_{0}, …, C_{k} satisfy the system of linear equations

[ \begin{matrix}z_{0} & z_{1} & ⋯ & z_{k} \\ z_{0}^{2} & z_{1}^{2} & ⋯ & z_{k}^{2} \\ ⋮ & ⋮ & ... & ⋮ \\ z_{0}^{k+1} & z_{1}^{k+1} & ⋯ & z_{k}^{k+1} \end{matrix} ] [ C_{0} ; C_{1} ; ⋮ ; C_{k} ] = [ 1 1 ⋮ 1 ] .

Let A be the square matrix on the left. By Cramer’s rule we have

C_{i} = \frac{1}{det A} | \begin{matrix}z_{0} & ⋯ & 1 & ⋯ & z_{k} \\ z_{0}^{2} & ⋯ & 1 & ⋯ & z_{k}^{2} \\ ⋮ & & ⋮ & & ⋮ \\ z_{0}^{k+1} & ⋯ & 1 & ⋯ & z_{k}^{k+1} \end{matrix} | .

The Vandermonde determinant in the numerator is not zero because the roots are all distinct and 1 is neither a root of p_{k} nor of P_{k}.

Finally, let z = x + i y ≠ φ_{k} be a root of p_{k} and denote its absolute value by r = |z| = \sqrt{x^{2} + y^{2}}. By Proposition 1.1, z is either the unique negative root (for odd k) or else y ≠ 0; in both cases x < r. Taking the absolute value of p_{k}(z) = 0 we have:

|z|^{k+1} = |z + 1| = \sqrt{(x + 1)^{2} + y^{2}} < \sqrt{x^{2} + y^{2} + 2r + 1}.

Equivalently, r^{k+1} < \sqrt{r^{2} + 2r + 1} = r + 1, i.e. p_{k}(r) < 0. The polynomial p_{k} is strictly increasing on [1, +∞) and p_{k}(φ_{k}) = 0. Therefore, p_{k}(x) > 0 for all x > φ_{k} and we conclude r < φ_{k}. Similarly, if z = x + i y ≠ ϕ_{k} is a root of P_{k}, we get

1 = |z| · |z − 1|^{k} = r\sqrt{r^{2} − 2x + 1}^{k} > r(r − 1)^{k} ⟹ P_{k}(r) < 0.

Again, P_{k}(x) > 0 for all x > ϕ_{k} and r < ϕ_{k} follows. This completes the proof. □

Corollary 2.2 lim_{n→∞} \frac{f_{n+k}^{(k)}}{f_{n}^{(k)}} = ϕ_{k}.

Proof. By the preceding theorem, consecutive ratios of the k-th lower Fibonacci sequences tend to φ_{k} so we have

\frac{f_{n+k}^{(k)}}{f_{n}^{(k)}} = \frac{f_{n+k}^{(k)}}{f_{n+k−1}^{(k)}} · \frac{f_{n+k−1}^{(k)}}{f_{n+k−2}^{(k)}} ⋯ \frac{f_{n+1}^{(k)}}{f_{n}^{(k)}} → φ_{k} · φ_{k} ⋯ φ_{k} = (φ_{k})^{k} = ϕ_{k}.

□

Just like ordinary Fibonacci numbers, their upper “cousins” can be expressed as sums of binomial coefficients. We will need the following lemma.

Lemma 2.3 For any k ≤ l ≤ m, ∑_{j=0}^{k}(−1)^{j}\binom{k}{j}\binom{m − j}{l} = \binom{m − k}{l − k}.

Proof. Let M be a set of m elements and suppose a subset of k elements is given. The right side enumerates all l-element subsets of M containing the given k elements. On the other hand, \binom{k}{j}\binom{m−j}{l} is the number of l-subsets avoiding at least j of the k given elements. The sum on the left equals the binomial coefficient on the right by inclusion-exclusion. □

Proposition 2.4 F_{n+1}^{(k)} = ∑_{i≥0} \binom{n − i}{k i}

Proof. Obviously, ∑_{i≥0} \binom{n−i}{k i} = 1 for all n ≤ k. The recurrence for the upper Fibonacci numbers can be rewritten as

∑_{j=0}^{k}(−1)^{j}\binom{k}{j}F_{n+k+1−j}^{(k)} = F_{n}^{(k)}.

By substituting appropriate sums of binomial coefficients we get

∑_{j=0}^{k}(−1)^{j}\binom{k}{j} ∑_{i≥0} \binom{n + k − j − i}{k i} = ∑_{i≥0} \binom{n − 1 − i}{k i} = ∑_{i≥1} \binom{n − i}{k(i − 1)}.

Equivalently,

∑_{i≥1} [∑_{j=0}^{k}(−1)^{j}\binom{k}{j}\binom{n + k − i − j}{k i} − \binom{n − i}{k(i − 1)}] = 0.

The terms in the square brackets are all zero by Lemma 2.3 for m = n+k −i and l = k i. Therefore, the considered sums satisfy the the initial conditions and the recurrence for the upper Fibonacci sequence. □

Members of the Lamé sequences can also be expressed as sums of binomial coefficients [11]:

L_{n+1}^{(k)} = ∑_{i=0}^{⌊n/k⌋} \binom{n − k i}{i}.

It would be of interest to find a similar formula for the lower Fibonacci sequences and to generalize other known properties of Fibonacci numbers.

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http://www.research.att.com/~njas/sequences/

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AMS Classification Number: 11B39