𝑘 -Step Sum and 𝑚 -Step Gap Fibonacci Sequence

Fortwogivenintegers k,m, weintroducethe k -stepsumand m -stepgapFibonaccisequencebypresentingarecurrenceformulathat generates the n th term as the sum of k successive previous terms starting the sum at the m th previous term. Known sequences, like Fibonacci, tribonacci, tetranacci, and Padovan sequences, are derived for specific values of k, m . Two limiting properties concerning the terms of the sequence are presented. The limits are related to the spectral radius of the associated {0,1} -matrix.


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.
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 {0, 1}-matrix.
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 ≥ 2, ≥ 1. One can write the following linear system, where (3) constitutes its first equation: Hence, using a ( + ) × 1 vector, the linear system in (8) can be formed as whereby it is obvious that the sequence ( ( , ) ) =1,2,... can be represented by a ( + ) × ( + ) matrix, , , which is a block matrix such that where the first row consists of the vector-matrices 1 , 2 ; the entries of the 1× vector 1 are equal to zero and the rest entries of the 1 × vector 2 are equal to one; the ( + − 1) × ( + − 1) matrix 3 is the identity matrix and the + − 1 entries of the ( + − 1) × 1 vector 4 are equal to zero.
Working as in the above, for ≥ 2, = 0, and using (4) with initial values in (5), we can write the following linear system: ] .
The × matrix, ,0 , of the coefficients of the above system, is defined as where the − 1 entries of the 1 × ( − 1) vector̃1 are equal to one, −1 is the ( − 1) × ( − 1) identity matrix, and the − 1 entries of the ( − 1) × 1 vector̃4 are equal to zero.

Remark 4. (i)
The well-known sequences, which are presented in Remark 2, correspond to ,0 in (12) for suitable integer value of ≥ 2 and = 0; (a) for = 2, the Fibonacci sequence corresponds to (b) for = 3, the tribonacci sequence corresponds to (c) for = 4, the tetranacci sequence corresponds to (iii) The matrix ,0 in (12) has been defined and the determinant of ,0 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]. (10) is given by Proof. The proof of (13) is based on the induction method. (13). Let be a fixed integer and assume that the formula in (13) is true for ; that is, Then, det( + +1 − +1, ) of the ( + + 1) × ( + + 1) matrix + +1 − +1, can be computed by using the Laplace expansion along the ( + +1)th column and the assumption of induction. Thus, we have   [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 −1, it is evident that Hence, , is a nonsingular and all the eigenvalues are nonzero.
Remark 6. Notice that, for = 0, (i) the th degree characteristic polynomial ( ) of the matrix ,0 in (12) is formulated by (13), which has presented in [9,10]; (ii) the authors in [10] have shown bounds for ( ,0 ); the lower bound is more accurate than the associated bound in (16); in particular, (iii) the determinant of ,0 is computed by (18) and derived the same result as in [6].
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 ( ( , ) ) =1,2,... , which depends on all the eigenvalues of , . Theorem 9. Let 1 , 2 , . . ., + −1 , ( , ) be the eigenvalues of , and the fixed integers , , with ≥ 2, ≥ 0. The th number of the sequence ( ( , ) ) =1,2,... is given by where , , for all = 1, 2, . . . , + − 1, are the determined coefficients of the partial-fraction decomposition in (25).
Proof. The inverse -transform on both sides of (25) for all = 1, 2, . . . yields The closed formula of in (26) follows from the above equation and the definitions of and Heaviside step functions.

Limiting Properties of -Step Sum and -
Step Gap Fibonacci Sequence The spectral radius of , in (10) is a characteristic quantity, which appears in (26) and for some cases of , is computed in Table 1. 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 (1, 2) verifying (16).
(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 1 , 2 = 1 , where = ( + − 1)/2.
(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 (1, 2) by (16) and has maximum modulus. The other real root is negative and lies in the interval [−1, 0) (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 = ( ( , )) + + −1 ( + −1 ) where = ( + − 2)/2.

Conclusions
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 {0, 1}-matrix. These limits can be regarded as the -step sum and -step gap Fibonacci sequence constants, like the tribonacci constant and the tetranacci constant.