Abstract

For two given integers , , we introduce the -step sumand -step gap Fibonacci sequence by presenting a recurrence formula that generates the th term as the sum of successive previous terms starting the sum at the th previous term. Known sequences, like Fibonacci, tribonacci, tetranacci, and Padovan sequences, are derived for specific values of , . Two limiting properties concerning the terms of the sequence are presented. The limits are related to the spectral radius of the associated -matrix.

1. Introduction

It is well-known that the Fibonacci sequence, the Lucas sequence, the Padovan sequence, the Perrin sequence, the tribonacci sequence, and the tetranacci sequence are very prominent examples of recursive sequences, which are defined as follows.

The Fibonacci numbers are derived by the recurrence relation , , with , [1], [2, A000045].

The Lucas numbers are derived by the recurrence relation , , with , and , [1], [2, A000032].

The Padovan numbers are derived by the recurrence relation , , with , , [2, A000931].

The Perrin numbers are derived by the recurrence relation , , with , , and , [2, A001608].

Both Fibonacci and Lucas numbers as well as both Padovan and Perrin numbers satisfy the same recurrence relation with different initial conditions.

Extending the above definitions, the -step Fibonacci sequences are derived [3]. For , the tribonacci numbers are derived by the recurrence relation , , with , and , [3–5], [2, A000073].

For , the tetranacci numbers are derived by the recurrence relation , , with , and , , [3], [2, A000078].

In this paper, we introduce -step sum and -step gap Fibonacci sequence, where the th term of the sequence is the sum of the successive previous terms starting at the th previous term, using 1’s as initial conditions. Further the closed formula of the th term of the sequence is given and the ratio of two successive terms tends to the spectral radius of the associated -matrix.

2. Definition of -Step Sum and -Step Gap Fibonacci Sequence

For the integers , , we define the -step sum and -step gap Fibonacci sequence , whose th term is given by the following recurrence relation: with Combining (1) and (2) notice that all the terms of the sequence are positive integers and is the sum of terms starting the sum at the th previous term from ; thus, (1) can be written equivalently as

Remark 1. (i) From (2)-(3) it is evident that for and all the terms of the sequence are equal to one. Hereafter consider , since the case is trivial.
(ii) For , (3) and (2) give the th term of the sequence , which is formulated as with initial values

Remark 2. The sequence gives known sequences for various values of the steps :(i)for , , (4)-(5) give the well-known Fibonacci sequence, ;(ii)for , , (4)-(5) give the tribonacci sequence, , [2, A000213];(iii)for , , (4)-(5) give the tetranacci sequence, , [2, A000288];(iv)for , , (2)-(3) give the Padovan sequence, , [2, A 000931].

In the following, the Dirac delta function (or function) is denoted by and the Heaviside step function (or the unit step function) .

Moreover, the th number of the sequence follows immediately from (2) and (3) using the above definition of the function and considering that the first negative indexed terms are equal to zero: which is formulated in the following proposition.

Proposition 3. For the given integers , , for all , the th number, , of the sequence is given by the following recurrence relation: with initial values as in (6).

In the following, we are going to demonstrate a close link between matrices and Fibonacci numbers in (3) with initial values in (2).

To this end, consider , . One can write the following linear system, where (3) constitutes its first equation: Hence, using a vector, the linear system in (8) can be formed as whereby it is obvious that the sequence can be represented by a matrix, , which is a block matrix such that where the first row consists of the vector-matrices , ; the entries of the vector are equal to zero and the rest entries of the vector are equal to one; the matrix is the identity matrix and the entries of the vector are equal to zero.

Working as in the above, for , , and using (4) with initial values in (5), we can write the following linear system: The matrix, , of the coefficients of the above system, is defined as where the entries of the vector are equal to one, is the identity matrix, and the entries of the vector are equal to zero.

Remark 4. (i) The well-known sequences, which are presented in Remark 2, correspond to in (12) for suitable integer value of and ; (a) for , the Fibonacci sequence corresponds to ;(b) for , the tribonacci sequence corresponds to ;(c) for , the tetranacci sequence corresponds to .
(ii) The Padovan sequence corresponds to the matrix by (10) with , .
(iii) The matrix in (12) has been defined and the determinant of has been investigated in [6] and some results on matrices related with Fibonacci numbers and Lucas numbers have been investigated in [7] and the transpose matrix of the general -matrix in [8].

Proposition 5. The th degree characteristic polynomial of in (10) is given by

Proof. The proof of (13) is based on the induction method. For , , the characteristic polynomial of is , which satisfies (13). Let be a fixed integer and assume that the formula in (13) is true for ; that is, Then, of the matrix can be computed by using the Laplace expansion along the th column and the assumption of induction. Thus, we have hence, (13) holds for , too. Thus the result follows by the induction method.

The set of all eigenvalues of is denoted by and called the spectrum of ; the nonnegative real number is called spectral radius of . Here, is an eigenvalue of , since the entries of are 0 or 1, [11, Theorem ]; further since where denotes the th entry of , [11, Theorem ], [12, Theorem 7], and [13].

Notice that if is an eigenvalue of , then , because has real coefficients. Further, since in (13) has the constant term equal to , it is evident that Hence, is a nonsingular and all the eigenvalues are nonzero.

Remark 6. Notice that, for ,(i)the th degree characteristic polynomial of the matrix in (12) is formulated by (13), which has presented in [9, 10];(ii)the authors in [10] have shown bounds for ; the lower bound is more accurate than the associated bound in (16); in particular, (iii)the determinant of is computed by (18) and derived the same result as in [6].

Example 7. Consider , , and the well-known Fibonacci sequence , as in Remark 2. According to Remark 4(i), the matrix is derived by (12). It is evident that the characteristic polynomial is given by and its roots are and , the well-known number as the golden ratio.

Example 8. Consider , . By (2)-(3) the associated sequence is formed as and , for all , which is well-known as the Padovan sequence (see, Remark 2). According to Remark 4(ii), the associated matrix is given by The characteristic polynomial is given by (13) as and its spectrum ,.

For the integers and , it is worth noting that, since the entries of the matrix are positive integers, is an irreducible matrix [11, Lemma ]; it follows that the spectral radius is a positive, simple (without multiplicity) eigenvalue of [11, Theorem 8.4.4]. In addition, the entries of are positive integers; thus is a primitive matrix [11, Corollary  8.5.9]; that is, is the unique eigenvalue with maximum modulus [11, Definition 8.5.0]. Hence, in the following, we denote , , ,, all the distinct eigenvalues of , for which the following inequality holds: Furthermore, rewriting (7) as the -transform on both sides of (23) yields From (24) it is worth noting that the poles of are the eigenvalues of , which are all simple (distinct) and the complex eigenvalues are conjugate; furthermore, the degrees of the polynomials of numerator and denominator of coincide. Thus, the partial-fraction decomposition of (24) is given by where , are real and the others coefficients are complex or real numbers.