1. Introduction
The Fibonacci sequence and the Lucas sequence are the two shining stars in the vast array of integer sequences. They have fascinated both amateurs and professional mathematicians for centuries. Also they continue to charm us with their beauty, their abundant applications, their ubiquitous habit of occurring in totally surprising and unrelated places. Fibonacci numbers have been generalized by different authors. Some authors have generalized the Fibonacci sequence by preserving the recurrence relation and altering the first two terms of the sequence ^{[10, 12, 17]}, while others have generalized the Fibonacci sequence by preserving the first two terms but altering the recurrence relation slightly ^{[14, 15, 18]}.
More recently, Fibonacci, Lucas, Pell, PellLucas, Modified Pell, Jacobhstal, JacobsthalLucas sequences were generalized for any positive real number k. Also the study of the kFibonacci sequence, the kLucas sequence, the kPell sequence, the kPellLucas sequence, the Modified kPell sequence, the kJacobhstal sequence and the kJacobsthal Lucas sequence appeared (see [16,8,13]). Falcon and plaza ^{[7]} has given several formulae for the sum of kFibonacci numbers with indexes in an arithmetic sequence. Falcon ^{[9]} defines the kLucas number with indexes in an arithmetic sequence. Also, deduced generating function and several sum formulae for these numbers with indexes in an arithmetic sequence.
1.1. k  Jacobsthal and k  Jacobsthal Lucas Numbers1.1.1. Definition 1.1 (kJacobsthal Numbers)For any positive real number k, the kJacobsthal sequence say ^{[13]} is defined recurrently by
 (1.1) 
with initial conditions.
Particular cases of definition (1.1)
If we obtain the Jacobsthal sequence ^{[11]} (A001045) ^{[16]}
If we obtain the sequence A002605 ^{[16]}.
First few terms of the kJacobsthal numbers (1.1) are
Some of the interesting properties that the kJacobsthal sequence satisfies are summarized as below ^{[13]}:
1.1.1.1. Binet’s Formula
The Binet formula for the n^{th} kJacobsthal numbers is
 (1.2) 
where , are the roots of the characteristic equation associated to the recurrence relation defined in equation (1.1) and ,, , .
1.1.1.2. Catalan’s Identity
 (1.3) 
1.1.1.3. D’Ocagne Identity
If m > n then
 (1.4) 
1.1.1.4. Convolution Product
 (1.5) 
Now, we introduce the kJacobsthal Lucas sequence, whose recurrence relation is the same as the kJacobsthal sequence.
1.2. Definition 1.2For any positive real number k, the kJacobsthalLucas sequence say ^{[2]} is defined recurrently by
 (1.6) 
with initial conditions .
As particular cases:
If we obtain the JacobsthalLucas sequence ^{[11]} (A014551) ^{[16]}.
If we obtain the sequence A080040 ^{[16]}.
First few terms of the kJacobsthal Lucas numbers equation (1.6) are
Some of the interesting properties that the kJacobsthal Lucas sequence satisfies are summarized as below ^{[2]}.
1.2.1. Binet’s FormulaThe Binet formula for the kJacobsthal numbers is
 (1.7) 
where , are the roots of the characteristic equation , , and associated to the recurrence relation defined in equation (1.6).
1.2.2. Catalan’s Identity  (1.8) 
1.2.3. D’Ocagne IdentityIf m > n then
 (1.9) 
Now, we prove some properties of the kJacobsthal numbers that we will be needed later.
 (1.10) 
Proof: Taking R.H.S. and applying the Binet’s formula for kJacobsthal numbers
That is
 (1.11) 
Proof: We will prove this by using the Mathematical induction method. For
we see that it is true for
Now for we have
again we see that it is true for
Now, suppose the formula is true until
 (1.12) 
Then,
That is .
2. On the kJacobsthal Numbers of Kind an+r
In this section, we shall derive some formulae for the sums of the k Jacobsthal numbers with index in an arithmetic sequence, say for fixed integer a and r such that Also, we have discuss generating function for these numbers with index in an arithmetic sequence.
First we prove following lemmas that will be needed later.
2.1. Lemma 2.1For all integers
 (2.1) 
Proof: Taking R.H.S. and applying the Binet’s formula for kJacobsthal numbers and .
That is
2.2. Lemma 2.2  (2.2) 
Proof: Taking R.H.S. and applying Binet’s formula and Lemma 2.1.
now, since , then the above formula can be rewritten as
equation (2.2) gives the general term of the kJacobsthal sequence as a linear combination of the two preceding terms.
2.2.1. Generating Function for the Sequence Suppose that be the generating function of the sequence with That is
 (2.3) 
Now multiplying both sides by the algebraic expression we get
Now, from equation (2.2), the summation of right hand side of the above equation vanishes. That is
 (2.4) 
from equation (1.10), we have
Hence equation (2.4) becomes
 (2.5) 
Particular cases:
For the different values of a and r, the generating function of the sequences are:
1) If and then,
which is the generating function of the kJacobsthal sequence ^{[13]}.
2) If , then
At ,
At ,
3) If , then
At ,
At ,
At ,
2.2.2. Sum of kJacobsthal numbers with arithmetic index In this section, we have discuss the sum formula for the kJacobsthal numbers with arithmetic index, where a and r are fixed integer such that
2.3. Theorem 2.3Sum of kJacobsthal number of kind is
 (2.6) 
Proof: Applying Binet’s formula for the k Jacobsthal numbers, we have
2.4. Corollary 2.4 Formula for sum of odd kJacobsthal numbers
If then equation (2.6) gives
 (2.7) 
For example: (1) If , then and, we have
(2) If, then, we have
If , then
If , then
If , then
(3) If , then , we have
If , then
If , then
If , then
If , then
If , then
2.5. Corollary 2.5Sum of even kJacobsthal numbers.
If , then equation (2.6) is
 (2.8) 
For example: (1) If, then , we have
If , then
If , then
(2) If , then , we have
If , then
If , then
If , then
If , then
Now, we have considered the alternating sequence . By the previous method we can also find the sum formula for this sequence.
2.6. Theorem 2.6Alternating sum of the kJacobsthal numbers with index is given by
 (2.9) 
Proof: Taking L.H.S. and applying the binet formula for the kJacobsthal numbers equation (1.2) we have
that is
For different values of a and r, the above sum can be written as
For we have
For,
a) If then
b) If then
For
If then
If then
If then
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