Research Article | Open Access
-Step Sum and -Step Gap Fibonacci Sequence
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.
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 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.
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).
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  and some results on matrices related with Fibonacci numbers and Lucas numbers have been investigated in  and the transpose matrix of the general -matrix in .
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 .
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  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 .
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.
In the following theorem, we are able to present the closed formula of the terms of the sequence , which depends on all the eigenvalues of .
Theorem 9. Let , , , , be the eigenvalues of and the fixed integers , , with , . The th number of the sequence is given by where , , for all , are the determined coefficients of the partial-fraction decomposition in (25).
3. Limiting Properties of -Step Sum and -Step Gap Fibonacci Sequence
From the values in Table 1 observe that the spectral radius (i)increases as increases and remains constant;(ii)decreases as increases and remains constant;(iii)lies in the interval verifying (16).Note that for , , the spectral radius is the tribonacci constant and for , , the spectral radius is the tetranacci constant .
The significance of is presented in the following theorem.
Proof. Consider that the polar form of the determined coefficients in (25) is denoted by , and the eigenvalues (except the spectral radius) , for all . The substitution of , from the polar forms in (26) yields
Using (30) and the property of the spectral radius from (22), we can write
Since and are bounded sequences as well as the inequality (22) implies for every , it is obvious that
Thus, the validity of (28) follows from (31) and (32).
Furthermore, it is well known that for a sequence of nonzero complex numbers, if , then [14, Chapter 1], whereby it is evident that for the sequence of the positive integers (), the equality (29) follows immediately from (28).
Remark 11. Notice that for every the formulas of in (26) and (30) are equivalent. Additionally, notice the following.(i)If is odd, then the characteristic polynomial in (13) has one real root, , and the others are complex conjugate. Thus, the complex eigenvalues and the coefficients in (25) appear in complex conjugate pairs, which are denoted by , , , , , , and , , , , , , , respectively. Then, using the complex conjugate properties, (30) follows where .(ii)If is even, then the characteristic polynomial in (13) has two real roots and the others are complex conjugate. The one real root is the unique real positive root ; it lies in the interval by (16) and has maximum modulus. The other real root is negative and lies in the interval (see in Acknowledgements). Thus, the complex eigenvalues and the coefficients in (25) appear in complex conjugate pairs and , are denoted as in (i). Then, using the complex conjugate properties, (30) follows where .
Example 12. Consider the Padovan sequence of the Example 8. Notice that , and is odd. The eigenvalues of are given in Example 8, , , and . Since , it is evident that the inequality (22) is verified. The partial-fraction decomposition as in (25) yields , +, and .
Thus, for the th number of the Padovan sequence is computed by (33) and given by Now, the limited properties of the Padovan sequence are derived by (28) and (29):
Example 13. Consider the 2-step sum and 2-step gap Fibonacci sequence. Notice that is even. The eigenvalues of are , +, , and . The partial-fraction decomposition as in (25) yields , + , , and . Thus, for the th number of the sequence is computed by (34) and given by Now, the limited properties of the sequence are derived by (28) and (29):
The -step sum and -step gap Fibonacci sequence was introduced. A recurrence formula was presented generating the th term of the sequence as the sum of successive previous terms starting the sum at the th previous term. It was noticed that known sequences, like Fibonacci, tribonacci, tetranacci, and Padovan sequences, are derived for specific values of . A closed formula of the th term of the sequence was given. The limiting properties concerning the ratio of two successive terms as well as the th root of the th term of the sequence were presented. It was shown that these two limits are equal to each other and are related to the spectral radius of the associated -matrix. These limits can be regarded as the -step sum and -step gap Fibonacci sequence constants, like the tribonacci constant and the tetranacci constant.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank Dr. Aristides Kechriniotis for his valuable comments about the roots of the characteristic polynomial in (13), verifying that the maximum modulus of the unique real positive root lies in the interval and that the second real root lies in the interval in the case where is even.
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