INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A32

MORE ON THE FIBONACCI SEQUENCE AND HESSENBERG MATRICES

Morteza Esmaeili

Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran

                          emorteza@cc.iut.ac.ir

Received: 7/7/06, Revised: 9/25/06, Accepted: 10/19/06, Published: 11/1/06

Abstract

Five new classes of Fibonacci-Hessenberg matrices are introduced. Further, we introduce the notion of two-dimensional Fibonacci arrays and show that three classes of previously known Fibonacci-Hessenberg matrices and their generalizations satisfy this property. Simple systems of linear equations are given whose solutions are Fibonacci fractions.

1. Introduction

The Fibonacci sequence is defined by f_{0} = 0, f_{1} = 1 and f_{n} = f_{n−1} + f_{n−2}, n ≥ 2. An n×n matrix A is called a (lower) Hessenberg matrix if all entries above the superdiagonal are zero. As an example set A_{1} = (1) and define A_{n} by:

A_{n} := ( \begin{matrix}2 & 1 & 0 & ⋯ & ⋯ & 0 \\ 1 & 2 & 1 & ... & ... & ⋮ \\ ⋮ & ... & ... & ... & ... & ⋮ \\ ⋮ & ... & ... & ... & ... & 0 \\ ⋮ & ... & ... & ... & 2 & 1 \\ 1 & ⋯ & ⋯ & ⋯ & 1 & 1 \end{matrix} )_{n×n.}

These matrices are Hessenberg and the determinant of A_{n} is the nth Fibonacci number f_{n}. Several Hessenberg matrices whose determinants are Fibonacci numbers have been introduced in [1], [2], [4], and [5]. Strang [5] has introduced real tridiagonal matrices such that their determinants are Fibonacci numbers, while in [2] we see complex Hessenberg matrices with this property. It has been shown in [4] that the maximum determinant achieved by n × n Hessenberg (0, 1)-matrices is the nth Fibonacci number f_{n} and a class of matrices (denoted in this paper by E_{n,0}) achieving this bound has been introduced.

In this paper, we consider sequences of Hessenberg matrices whose determinants are in the form tf_{n−1} +f_{n−2} or f_{n−1} +tf_{n−2} for some real or complex number t. Such matrices will be referred to as Fibonacci-Hessenberg matrices.

In Section 2 we introduce five new classes of Fibonacci-Hessenberg matrices. As a new concept, the two-dimensional Fibonacci array is introduced in Section 3. Three classes of Fibonacci-Hessenberg matrices satisfying this property are given.

2. More Fibonacci-Hessenberg Matrices

As mentioned above, several connections between the Fibonacci sequence and Hessenberg matrices have been given in [1], [2], [4], and [5]. In this section we develop some of these connections and provide more examples.

The Fibonacci recurrence relation a_{n} = a_{n−1} + a_{n−2} beginning with a_{1} = 1 and a_{2} = t produces the sequence 1, t, t + 1, 2t + 1, 3t + 2, 5t + 3, · · ·. Thus a_{n} = tf_{n−1} + f_{n−2} for n ≥ 1 and a_{n} = f_{n} if and only if t = 1. On the other hand, the sequence a_{n} = a_{n−1} +a_{n−2} starting at a_{1} = t and a_{2} = 1 satisfies a_{n} = tf_{n−2} + f_{n−1}.

Definition 1 Given a real or complex number t and an integer n, we refer to numbers tf_{n−1} + f_{n−2} and f_{n−1} + tf_{n−2} as type 1 and type 2, respectively, (t, n)-Fibonacci, briefly t-Fibonacci, numbers. A sequence of Hessenberg matrices A_{1}, A_{2}, A_{3}, ⋯, where A_{n} is an n × n matrix, is defined to be a Fibonacci-Hessenberg matrix if there exists an integer m > 0 and a number t such that, for each n ≥ m, the determinant of A_{n} is a t-Fibonacci number and such that the determinants are of the same type.

Example 1 Given a number t, let R_{n,t} denote the n × n matrix given below.

R_{n,t} := ( \begin{matrix}2 & 1 & 0 & ⋯ & ⋯ & 0 \\ 1 & 2 & 1 & ... & ... & ⋮ \\ ⋮ & ... & ... & ... & ... & ⋮ \\ ⋮ & ... & ... & ... & ... & 0 \\ ⋮ & ... & ... & ... & 2 & 1 \\ 1 & ⋯ & ⋯ & ⋯ & 1 & t + 1 \end{matrix} )_{n×n.}

The determinant of R_{n,t} is denoted by r_{n,t}. It can be shown that r_{n,t} = tf_{n+1} +f_{n}, n ≥ 1, (see relation (10) in the proof of Theorem 2) and hence R_{n,t} is a Fibonacci-Hessenberg matrix. Thus, for instance, r_{n,−1} = −f_{n+1} + f_{n} = −f_{n−1}; that is r_{n,−1} generates the additive inverse of the Fibonacci sequence. The Lucas numbers are defined by l_{1} = 1, l_{2} = 3 and l_{n} = l_{n−1} + l_{n−2} for n > 2. One can easily verify by induction that l_{n} = f_{n−1} + f_{n+1}. Hence r_{n,3} = 3f_{n+1} + f_{n} = f_{n+1} + f_{n+3} = l_{n+2}. For t = 0 we have r_{n,0} = f_{n},

Table 1: Determinant r_{n,t} for 1 ≤ n ≤ 5 and t = −1, 0, 1, 2, 3.

\begin{matrix}t\n & 1 & 2 & 3 & 4 & 5 \\ 3 & 4 & 7 & 11 & 18 & 29 \\ 2 & 3 & 5 & 8 & 13 & 21 \\ 1 & 2 & 3 & 5 & 8 & 13 \\ 0 & 1 & 1 & 2 & 3 & 5 \\ −1 & 0 & −1 & −1 & −2 & −3 \end{matrix}

while r_{n,1} = f_{n+1} + f_{n} = f_{n+2} and r_{n,2} = 2f_{n+1} + f_{n} = f_{n+3}. These are illustrated by Table 1.

Given a positive integer n, let C_{n,t} be the n × n matrix in which the entries below the diagonal are 1, the lowest entry of the nth column is t + 1 and the other diagonal entries are 2, the entries on the superdiagonal are −1 and the entries above the superdiagonal are zero. Changing the first element of the first column in C_{n,t} to 2, we get another Hessenberg matrix denoted by B_{n,t}. Matrices C_{5,t} and B_{5,t} are given below.

C_{5,t} := ( \begin{matrix}2 & −1 & 0 & 0 & 0 \\ 1 & 2 & −1 & 0 & 0 \\ 1 & 1 & 2 & −1 & 0 \\ 1 & 1 & 1 & 2 & −1 \\ 1 & 1 & 1 & 1 & t + 1 \end{matrix} ) B_{5,t} := ( \begin{matrix}1 & −1 & 0 & 0 & 0 \\ 1 & 2 & −1 & 0 & 0 \\ 1 & 1 & 2 & −1 & 0 \\ 1 & 1 & 1 & 2 & −1 \\ 1 & 1 & 1 & 1 & t + 1 \end{matrix} )

Proposition 1 The determinant of C_{n,t}, denoted c_{n,t}, is c_{n,t} = f_{2n} + tf_{2n−1}, n ≥ 1, and B_{n,t} has determinant b_{n,t} = f_{2n−1} + tf_{2n−2}, n ≥ 1.

Proof. We prove the statements by induction on n. It is obvious that these statements hold for n = 1, 2. Suppose n ≥ 3. By the cofactor expansion along the first row we get c_{n,t} = 2c_{n−1,t} + b_{n−1,t} and b_{n,t} = c_{n−1,t} + b_{n−1,t}. It follows from these two relations that

b_{n,t} = c_{n,t} − c_{n−1,t}.  (1)

Relations (1) and c_{n,t} = 2c_{n−1,t} + b_{n−1,t} imply that c_{n,t} = 3c_{n−1,t} − c_{n−2,t}. Using the induction hypothesis we get

c_{n,t} = 3c_{n−1,t} − c_{n−2,t}
= 3(f_{2n−2} + tf_{2n−3}) − (f_{2n−4} + tf_{2n−5})
= (3f_{2n−2} − f_{2n−4}) + t(3f_{2n−3} − f_{2n−5})
= f_{2n} + tf_{2n−1}.

Finally, it follows from b_{n,t} = c_{n,t} − c_{n−1,t} and c_{n,t} = f_{2n} + tf_{2n−1} that b_{n,t} = f_{2n−1} + tf_{2n−2}. □

The three classes of Fibonacci-Hessenberg matrices given above are generalizations of matrices D_{n}, C_{n}, and B_{n} introduced in [1]. In fact the matrices D_{n}, C_{n}, and B_{n} given in [1] are R_{n,1}, C_{n,1}, and B_{n,1}, respectively.

Now we introduce five new classes of Fibonacci-Hessenberg matrices. Given a number t, let K_{n,t} be the n × n Hessenberg matrix in which the superdiagonal entries are −1, the entry located on the nth row and nth column is t + 1 and the other diagonal entries are 2, and the entries on each column and below the diagonal are alternately −1 and 1 starting with −1. The matrix K_{5,t} is given by (2).

Replacing the top-left entry (the entry located in the first row and first column) in K_{n,t} with 1, we obtain another Hessenberg matrix denoted by L_{n,t}. Replacing the superdiagonal entries in both K_{n,t} and L_{n,t} with 1, two more classes of Hessenberg matrices, denoted \underline{K}_{n,t} and \underline{L}_{n,t}, respectively, are obtained.

\begin{matrix}K_{5,t} = ( \begin{matrix}2 & −1 & 0 & 0 & 0 \\ −1 & 2 & −1 & 0 & 0 \\ 1 & −1 & 2 & −1 & 0 \\ −1 & 1 & −1 & 2 & −1 \\ 1 & −1 & 1 & −1 & t + 1 \end{matrix} ) L_{5,t} = ( \begin{matrix}1 & −1 & 0 & 0 & 0 \\ −1 & 2 & −1 & 0 & 0 \\ 1 & −1 & 2 & −1 & 0 \\ −1 & 1 & −1 & 2 & −1 \\ 1 & −1 & 1 & −1 & t + 1 \end{matrix} ) \\ \underline{K}_{5,t} = ( \begin{matrix}2 & 1 & 0 & 0 & 0 \\ −1 & 2 & 1 & 0 & 0 \\ 1 & −1 & 2 & 1 & 0 \\ −1 & 1 & −1 & 2 & 1 \\ 1 & −1 & 1 & −1 & t + 1 \end{matrix} ) \underline{L}_{5,t} = ( \begin{matrix}1 & 1 & 0 & 0 & 0 \\ −1 & 2 & 1 & 0 & 0 \\ 1 & −1 & 2 & 1 & 0 \\ −1 & 1 & −1 & 2 & 1 \\ 1 & −1 & 1 & −1 & t + 1 \end{matrix} ) \end{matrix}  (2)

Theorem 1 Let k_{n,t}, l_{n,t}, \underline{k}_{n,t} and \underline{l}_{n,t} denote the determinants of K_{n,t}, L_{n,t}, \underline{K}_{n,t} and \underline{L}_{n,t}, respectively. Then

\{ \begin{matrix}& k_{n,t} = f_{n} + tf_{n+1}, n ≥ 1; \\ \{ & \begin{matrix}l_{1,t} = t + 1, \\ l_{n,t} = k_{n−2,t} = f_{n−2} + tf_{n−1}, n ≥ 2; \end{matrix} \\ & \underline{k}_{n,t} = f_{2n} + tf_{2n−1}, n ≥ 1; \\ \{ & \begin{matrix}\underline{l}_{1,t} = 1 + t, \\ \underline{l}_{n,t} = \underline{k}_{n−1,t} + \underline{l}_{n−1,t} = f_{2n−1} + tf_{2n−2}, n ≥ 2. \end{matrix} \end{matrix}

Therefore, the four introduced classes of Hessenberg matrices are indeed Fibonacci-Hessenberg matrices.

Proof. The proof is by induction on n. Due to the similarity between matrices K_{n,t} (L_{n,t}) and \underline{K}_{n,t} (resp. \underline{L}_{n,t}), we just prove the first two statements. It is easily verified that the statements hold for 1 ≤ n ≤ 3. Assume that n ≥ 4. Using cofactor expansion along the first row we obtain:

\{ \begin{matrix}l_{n,t} = k_{n−1,t} − l_{n−1,t}, & n ≥ 4; \\ k_{n,t} = 2k_{n−1,t} − l_{n−1,t}, & n ≥ 4. \end{matrix}  (3)

Therefore,

\begin{aligned}l_{n,t} & = k_{n−1,t} − l_{n−1,t} \\ & = (2k_{n−2,t} − l_{n−2,t}) − (k_{n−2,t} − l_{n−2,t}) \\ & = k_{n−2,t}. \end{aligned}  (4)

Relations (3) and (4) imply

k_{n,t} = 2k_{n−1,t} − l_{n−1,t} = 2k_{n−1,t} − k_{n−3,t}, n ≥ 4.  (5)

This, together with the induction hypothesis, result in:

k_{n,t} = 2k_{n−1,t} − k_{n−3,t} = 2 (f_{n−1} + tf_{n}) − (f_{n−3} + tf_{n−2})
= (f_{n−1} + f_{n−1} − f_{n−3}) + (tf_{n} + tf_{n} − tf_{n−2})
= (f_{n−1} + f_{n−2} + f_{n−3} − f_{n−3}) + (tf_{n} + tf_{n−1} + tf_{n−2} − tf_{n−2})

= f_{n} + tf_{n+1}. □

Define E_{1,t} = (t + 1) and E_{2,t} = ( \begin{matrix}1 & 1 \\ 0 & t + 1 \end{matrix} ). Given the n × n matrix E_{n,t}, a new matrix E_{n+1,t} is formed by adding one row of weight one and starting with 1 to the top of E_{n,t} and then adding a new column with alternating 1’s and 0’s, starting with a 1, to the left of the obtained matrix. The matrix E_{5,t} is given below. The matrix E_{n,0} was introduced in [4] and it was shown in [4] that the determinant of E_{n,0}, n ≥ 1, is f_{n}. Let i denote the usual complex unit with i^{2} = −1. Replacing the entry of E_{n,t} located in the ith row and (i + 1)th column, 1 ≤ i < n, with (−1)^{i+n}i, we obtain another Hessenberg matrix denoted by H_{n,t}. The matrix H_{5,t} is shown below.

E_{5,t} = ( \begin{matrix}1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 1 & 0 & t + 1 \end{matrix} ) H_{5,t} = ( \begin{matrix}1 & i & 0 & 0 & 0 \\ 0 & 1 & −i & 0 & 0 \\ 1 & 0 & 1 & i & 0 \\ 0 & 1 & 0 & 1 & −i \\ 1 & 0 & 1 & 0 & t + 1 \end{matrix} )

Proposition 2 The determinants of E_{n,t} and H_{n,t}, denoted by e_{n,t} and h_{n,t}, respectively, are e_{1,t} = h_{1,t} = t + 1 and e_{n,t} = h_{n,t} = tf_{n−1} + f_{n} for n ≥ 2. Thus E_{n,t} and H_{n,t} are Fibonacci-Hessenberg matrices.

Proof. Consider H_{n,t}. Obviously we have h_{1,t} = h_{2,t} = t + 1 and h_{3,t} = 2t + 3. Suppose n ≥ 4 and that the statement holds for each integer 0 < m < n. Let H′_{n,t} be the matrix obtained from H_{n,t} by means of deleting the first row and second column. It is easy to see that using cofactor expansion along the first row on both H_{n,t} and H′_{n,t} results in h_{n,t} = h_{n−1,t} + h_{n−2,t}. Therefore, it follows from the induction hypothesis that h_{n,t} = h_{n−1,t} + h_{n−2,t} = (tf_{n−2} + f_{n−1}) + (tf_{n−3} + f_{n−2}) = tf_{n−1} + f_{n}. The same argument applies to E_{n,t}. □

3. Two-dimensional Fibonacci Arrays

From among the introduced Fibonacci matrices, the matrices R_{n,t}, C_{n,t} and E_{n,t} have a further interesting property. Given an n × n matrix M, let M^{(i)} be the matrix obtained from M by replacing its ith column with the all one column vector 1. The mentioned matrices have the property that the determinants of their associated matrices R_{n,t}^{(i)}, C_{n,t}^{(i)} and E_{n,t}^{(i)}, 1 ≤ i ≤ n, are t-Fibonacci numbers. This leads to a connection between Fibonacci fractions and the all one vector 1.

Theorem 2 Let r_{n,t}^{(i)}, c_{n,t}^{(i)} and e_{n,t}^{(i)} be determinants of R_{n,t}^{(i)}, C_{n,t}^{(i)} and E_{n,t}^{(i)}, respectively. Then we have

\{ \begin{matrix}r_{n,t}^{(i)} = tf_{n−i} + f_{n−i−1}, n ≥ i ≥ 1; \\ r_{n,t} = t + ∑_{i=1}^{n} r_{n,t}^{(i)}; \\ c_{n,t}^{(i)} = f_{2(n−i)+1} + tf_{2(n−i)}, n ≥ i ≥ 1; \\ c_{n,t} = t + ∑_{i=1}^{n} c_{n,t}^{(i)}; \\ e_{1,t}^{(1)} = 1; e_{n,t}^{(1)} = tf_{n−3} + f_{n−2}, n ≥ 2; \\ e_{n,t}^{(i)} = tf_{n−i} + f_{n−i+1}, n ≥ i ≥ 2; \\ 2e_{n,t} = (t + 1) + ∑_{i=1}^{n} e_{n,t}^{(i)}, n ≥ 2. \end{matrix}  (6)

Proof. We use induction on n to prove the statements related to the Fibonacci-Hessenberg matrices R_{n,t}; similar arguments apply to the other two classes of matrices. Setting f_{0} = 0 and f_{−1} = 1, it is easy to verify that the statements hold for 1 ≤ n ≤ 3. Evaluating the determinants of matrices R_{n,t} and R_{n,t}^{(1)} by cofactor expansion along the first row, we have

\{ \begin{matrix}r_{n,t}^{(1)} = r_{n−1,t} − r_{n−1,t}^{(1)}, & n ≥ 3; \\ r_{n−1,t} = 2r_{n−2,t} − r_{n−2,t}^{(1)}, & n ≥ 3. \end{matrix}  (7)

Therefore,

\begin{aligned}r_{n,t}^{(1)} & = r_{n−1,t} − r_{n−1,t}^{(1)} \\ & = (2r_{n−2,t} − r_{n−2,t}^{(1)}) − (r_{n−2,t} − r_{n−2,t}^{(1)}) \\ & = r_{n−2,t}. \end{aligned}  (8)

It follows from (7) and (8) that

r_{n,t} = 2r_{n−1,t} − r_{n−1,t}^{(1)} = 2r_{n−1,t} − r_{n−3,t}, n ≥ 3.  (9)

This together with the induction hypothesis result in

\begin{aligned}r_{n,t} & = 2r_{n−1,t} − r_{n−3,t} \\ & = 2 (tf_{n} + f_{n−1}) − (tf_{n−2} + f_{n−3}) \\ & = t(2f_{n} − f_{n−2}) + (2f_{n−1} − f_{n−3}) \\ & = tf_{n+1} + f_{n}, \end{aligned}  (10)

and hence

r_{n,t}^{(1)} = r_{n−2,t} = tf_{n−1} + f_{n−2}.  (11)

By cofactor expansion along the first row, one can easily verify that r_{n,t}^{(i)} = r_{n−1,t}^{(i−1)} if 2 ≤ i ≤ n − 1, and thus

r_{n,t}^{(i)} = r_{n−i−1,t} = tf_{n−i} + f_{n−i−1}, 2 ≤ i ≤ n − 1.  (12)

The augmented matrices obtained from a given matrix M by adding the all one column vector 1 to the left and the right of M are denoted by 1M and M1, respectively. Evaluating the determinants by using cofactors along the first row, we obtain r_{n,t}^{(n)} = 2r_{n−1,t}^{(n−1)} + (−1)^{n+1}det(1M) where M1 is the matrix R_{n−1,t}^{(n−1)}. Therefore,

r_{n,t}^{(n)} = 2r_{n−1,t}^{(n−1)} + (−1)^{2n−1}r_{n−1,t}^{(n−1)} = r_{n−1,t}^{(n−1)} = 1 = tf_{0} + f_{−1}.  (13)

It is easily checked by induction on n that ∑_{j=−1}^{n} f_{j} = f_{n+2}. This together with the equations r_{n,t} = tf_{n+1} + f_{n} and r_{n,t}^{(i)} = tf_{n−i} + f_{n−i−1} imply that r_{n,t} = t + ∑_{i=1}^{n} r_{n,t}^{(i)}. □

Corollary 1 (Fibonacci Fractions and Hessenberg Matrices) The system of equations R_{n,t}x = 1 has the unique solution

x_{i} = \frac{f_{n−i−1} + tf_{n−i}}{f_{n} + tf_{n+1}} , 1 ≤ i ≤ n.  (14)

Similarly, the system of equations C_{n,t}x = 1 has the solution

x_{i} = \frac{f_{2(n−i)+1} + tf_{2(n−i)}}{f_{2n} + tf_{2n−1}} , 1 ≤ i ≤ n.  (15)

We also have x_{1} = \frac{tf_{n−3}+f_{n−2}}{tf_{n−1}+f_{n}} and x_{i} = \frac{tf_{n−i}+f_{n−i+1}}{tf_{n−1}+f_{n}}, 2 ≤ i ≤ n, as the unique solution of E_{n,t}x = 1.

Proof. It follows from (6) and Cramer’s rule that the system R_{n,t}x = 1 has unique

solution x_{i} = \frac{r_{n,t}^{(i)}}{r_{n,t}} = \frac{f_{n−i−1}+tf_{n−i}}{f_{n}+tf_{n+1}} , 1 ≤ i ≤ n. The same argument applies to the systems C_{n,t}x = 1 and E_{n,t}x = 1. □

In particular, according to (14), for t = 0, 1, 2 the system R_{n,t}x = 1 has solutions:

\{ \begin{matrix}x_{i} = \frac{f_{n−i−1}}{f_{n}} , & t = 0; \\ x_{i} = \frac{f_{n−i}+f_{n−i−1}}{f_{n+1}+f_{n}} = \frac{f_{n−i+1}}{f_{n+2}} , & t = 1; \\ x_{i} = \frac{2f_{n−i}+f_{n−i−1}}{2f_{n+1}+f_{n}} = \frac{f_{n−i+2}}{f_{n+3}} , & t = 2. \end{matrix}

It also follows from (15) that for t = 0, 1, 2 the system C_{n,t}x = 1 has solutions

\{ \begin{matrix}x_{i} = \frac{f_{2(n−i)+1}}{f_{2n}} , & t = 0; \\ x_{i} = \frac{f_{2(n−i+1)}}{f_{2n+1}} , & t = 1; \\ x_{i} = \frac{f_{2(n−i+1)}+f_{2(n−i)}}{f_{2n+1}+f_{2n−1}} , & t = 2. \end{matrix}

Consider the n × n Fibonacci-Hessenberg matrix R_{n,t}. For 1 ≤ i ≤ n we have r_{n,t}^{(i)} = tf_{n−i} + f_{n−i+1}. This t-Fibonacci number depends on both n and i, and hence we have a two-dimensional array r(n, i; t) of t-Fibonacci numbers.

Table 2: The Values of r_{n,t}^{(i)} for 1 ≤ n, i ≤ 6.
n\i  1  2  3  4  5  6
1  tf_{0} + f_{−1}
2  tf_{1} + f_{0} tf_{0} + f_{−1}
3  tf_{2} + f_{1}  tf_{1} + f_{0} tf_{0} + f_{−1}
4  tf_{3} + f_{2}  tf_{2} + f_{1}  tf_{1} + f_{0} tf_{0} + f_{−1}
5  tf_{4} + f_{3}  tf_{3} + f_{2}  tf_{2} + f_{1}  tf_{1} + f_{0} tf_{0} + f_{−1}
6  tf_{5} + f_{4}  tf_{4} + f_{3}  tf_{3} + f_{2}  tf_{2} + f_{1}  tf_{1} + f_{0} tf_{0} + f_{−1}

Table 3: The Values of r_{n,1}^{(i)} for 1 ≤ n, i ≤ 6.
n\i 1 2 3 4 5 6
1  1
2  1 1
3  2 1 1
4  3 2 1 1
5  5 3 2 1 1
6  8 5 3 2 1 1

Table 2 represents r(n, i; t) for 1 ≤ n, i ≤ 6. Asymptotically, all rows and columns of the array r(n, i; t) are the same. Table 3 represents r(n, i; 1) for 1 ≤ n, i ≤ 6. For a fixed n the nth row of the array consists of the first n Fibonacci numbers and, for each i, the ith column, starting at the ith entry, is also the Fibonacci sequence.

In the context of systems theory [3], we may consider the determinant function as an operator and interpret r(n, i; t) = Det(R_{n,t}^{(i)}) as a two-dimensional system. The results show that this system is invariant in the sense that its output is always a t-Fibonacci number. We can also say that the system is invariant with respect to fixing any of the two variables n and i; that is, its output with a fixed n is identical to the output when i is fixed and n varies.

References

[1] N.D. Cahill, J.R. D′Errico, D.A. Narayan and J.Y. Narayan, “Fibonacci Determinants,” The College Math. Journal of The Math. Association of America, vol. 33, no. 3, pp. 221–225, May 2002.

[2] N.D. Cahill, J.R. D′Errico, J. Spence, “Complex factorizations of the Fibonacci and Lucas numbers,” Fibonacci Quarterly, vol. 41, no. 1, pp. 13–19, Feb. 2003.

[3] K.B. Datta and B. M. Mohan, Orthogonal Functions in Systems and Control, IIT, Kharagpur, India, 1995.

[4] Ching Li,“The Maximum Determinant of an nxn Lower Hessenberg (0,1) matrix,” Linear Algebra and Its Applications, 183, pp. 147–153, 1993.

[5] G. Strang, Introduction to Linear Algebra, 2nd Edition, Wellesley MA, Wellesley-Cambridge, 1998.