Keywords: kFibonacci numbers, kFibonacciLike numbers, Binet’s formula
Turkish Journal of Analysis and Number Theory, 2014 2 (1),
pp 912.
DOI: 10.12691/tjant213
Received December 20, 2013; Revised January 28, 2014; Accepted February 11, 2014
Copyright © 2014 Science and Education Publishing. All Rights Reserved.
1. Introduction
Many authors have generalized second order recurrence sequences by preserving the recurrence relation and alternating the first two terms of the sequence and some authors have generalized these sequences by preserving the first two terms of the sequence but altering the recurrence relation slightly.
Kalman and Mena ^{[7]} generalize the Fibonacci sequence by
 (1.1) 
Horadam ^{[2]} defined generalized Fibonacci sequence by
 (1.2) 
where p and q are arbitrary integers.
Singh, Sikhwal, and Bhatnagar ^{[5]}, defined FibonacciLike sequence by recurrence relation
 (1.3) 
The associated initial conditions and are the sum of the Fibonacci and Lucas sequences respectively, i.e. and .
Natividad ^{[9]}, Deriving a Formula in solving FibonacciLike sequence. He found missing terms in FibonacciLike sequence and solved by standard formula.
Gupta, Panwar and Sikhwal ^{[19]}, defined generalized Fibonacci sequences and derived its identities connection formulae and other results. Gupta, Panwar and N. Gupta ^{[18]}, stated and derived identities for FibonacciLike sequence. Also described and derived connection formulae and negation formula for FibonacciLike sequence. Singh, Gupta and Panwar ^{[6]}, present many Combinations of Higher Powers of FibonacciLike sequence.
The kFibonacci numbers defined by Falco’n and Plaza ^{[13]}, depending only on one integer parameter k as follows, For any positive real number k, the kFibonacci sequence is defined recurrently by
 (1.4) 
Many of the properties of these sequences are proved by simple matrix algebra. This study has been motivated by the arising of two complex valued maps to represent the two antecedents in a specific fourtriangle partition. In ^{[15]}, Falcon and Plaza k Fibonacci sequence generalizes, between others, both the classical Fibonacci sequence and the Pell sequence. In this paper many properties of these numbers are deduced and related with the socalled Pascal 2triangle. New generalized kFibonacci sequences have been introduced and studied. Several properties of these numbers are deduced and related with the socalled Pascal 2triangle. In addition, the generating functions for these kFibonacci sequences have been given.
In this paper, we introduced the kFibonacciLike sequence. Also we establish some of the interesting properties of kFibonacciLike numbers like Catalan’s identity, Cassini’s identity, d’ocagnes’s Identity, Binet’s formula and Generating function.
2. The kFibonacciLike Sequence
Definition: For any positive real number k, the kFibonacciLike sequence is defined by for ,
 (2.1) 
The first few kFibonacciLike numbers are
Particular case of k− generalized Fibonacci number is
and
3. Properties of kFibonacciLike Numbers
3.1. First Explicit Formula for kFibonacciLike Numbers In the 19th century, the French mathematician Binet devised two remarkable analytical formulas for the Fibonacci and Lucas numbers. In our case, Binet’s formula allows us to express the kFibonacciLike numbers in function of the roots of the following characteristic equation, associated to the recurrence relation (2.1)
 (3.1) 
Theorem 1: (Binet’s formula). The nth kFibonacciLike number is given by
 (3.2) 
where are the roots of the characteristic equation (3.1) and .
Proof: We use the Principle of Mathematical Induction (PMI) on n. It is clear the result is true for by hypothesis. Assume that it is true for such that, then
It follows from definition of kgeneralized Fibonacci numbers (2.1) and equation (3.2)
Thus, the formula is true for any positive integer n.
where and.
This completes the proof.
3.2. Catalan's IdentityCatalan's identity for Fibonacci numbers was found in 1879 by Eugene Charles Catalan a Belgian mathematician who worked for the Belgian Academy of Science in the field of number theory.
Theorem 2: (Catalan’s identity)
 (3.3) 
Proof: By Binet’s formula (3.2), we have
This completes the Proof.
3.3. Cassini's IdentityThis is one of the oldest identities involving the Fibonacci numbers. It was discovered in 1680 by JeanDominique Cassini a French astronomer.
Theorem 3: (Cassini’s identity or Simpson’s identity)
 (3.4) 
Proof. Taking in Catalan’s identity (3.3) the proof is completed.
In a similar way that before the following identity is proven:
3.4. d'Ocagne's IdentityTheorem 4: (d’ocagnes’s Identity) If then
 (3.5) 
3.5. Limit of the Quotient of Two Consecutive TermsA useful property in these sequences is that the limit of the quotient of two consecutive terms is equal to the positive root of the corresponding characteristic equation
Theorem 5:
 (3.6) 
Proof. By Binet’s formula (3.2), we have
and taking into account that , since, Eq. (3.6) is obtained.
Theorem 6: If , then
 (3.7) 
Proof. By Binet’s formula (3.2), we have
This completes the Proof.
Proposition 7: For any integer ,
 (3.8) 
Theorem 8: For any integer,
 (3.9) 
Proof . By Binet’s formula (3.2), we have
By summing up the geometric partial sums for . We obtain
This completes the Proof.
3.6. Generating Function for kFibonacciLike Sequence:Generating functions provide a powerful technique for solving linear homogeneous recurrence relations. Even though generating functions are typically used in conjunction with linear recurrence relations with constant coefficients, we will systematically make use of them for linear recurrence relations with non constant coefficients. In this paragraph, the generating function for kFibonacciLike sequence is given. As a result, k FibonacciLike sequence is seen as the coefficients of the corresponding generating function. Function defined in such a way is called the generating function of the k FibonacciLike sequence. So,
and then,
 (3.10) 
4. Conclusion
In this paper, kFibonacci pattern based sequence introduced which is known as kFibonacciLike sequence.
Many of the properties of this sequence are proved by simple algebra and Binet’s formula. Finally we present properties like Catalan’s identity, Cassini’s identity or Simpson’s identity and d’ocagnes’s identity for kFibonacciLike numbers. Further generating function of kFibonacciLike sequence is presented.
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