The generalized order-k Jacobsthal numbers - Hikari

Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 34, 1685 - 1694

The Generalized Order-k Jacobsthal Numbers Fatih Yilmaz and Durmus Bozkurt Department of Mathematics, Faculty of Science Selcuk University, 42003 Konya, Turkey [email protected] [email protected] Abstract In this paper, we consider the usual Jacobsthal numbers and defined a new sequence which is called generalized order-k Jacobsthal sequence. Then, we investigate some properties of the sequence, obtain generalized Binet formula and give a formula for sums of the Jacobsthal numbers.

Mathematics Subject Classification: 15A15; 11C20; 11B37; 05A15 Keywords: Jacobsthal Numbers; Sums; Binet formula; Matrix method

1

Introduction

It is known that the Jacobsthal sequence is defined by the following equation, for n ≥ 2 Jn = Jn−1 + 2Jn−2 , where J0 = 0 and J1 = 1. The Jacobsthal sequence is a special case of a sequence which is defined as a linear combination by Kalman, as following an+k = c1 an+k−1 + c2 an+k−2 + · · · + ck an , where c1 , c2 , ..., ck are real constants. Kalman [1] showed that number sequences can be derived by a matrix representation. He derived closed-form

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F. Yilmaz and D. Bozkurt

formulas for the generalized sequence by companion matrix method as follows: ⎡ ⎢ ⎢ ⎢ Ak = ⎢ ⎢ ⎣

⎤ c1 c2 ... ck−1 ck 1 0 ... 0 0 ⎥ ⎥ 0 1 ... 0 0 ⎥ ⎥. .. .. . . .. .. ⎥ . . . . . ⎦ 0 0 ... 1 0

Then by an inductive argument, he obtained ⎡ ⎤ ⎡ a0 an ⎢ a1 ⎥ ⎢ an+1 ⎢ ⎥ ⎢ Ank ⎢ . ⎥ = ⎢ .. . ⎣ . ⎦ ⎣ . ak−1

⎤ ⎥ ⎥ ⎥, ⎦

an+k−1

where an is the nth term of the sequence. In [3, 4] authors investigated some properties involving Jacobsthal numbers. Also in [4], authors gave matrix method for generating Jacobsthal sequence as following: F =

1 2 1 0

.

By taking positive integer powers of this matrix, it can be easily obtain that Jn+1 2Jn n . F = Jn 2Jn−1 In [5], Tascı and Kılıc defined order-k Lucas sequence in matrix representation by employing the matrix methods of Kalman. Also, authors give generalized Binet formula and sums of the generalized order-k Pell numbers [2]. At the present paper, we give a new generalization of the Jacobsthal numbers in matrix representation and obtain some properties of the sequence by matrix methods.

2

The Main Resuts

Define k-sequences of the generalized order-k Jacobsthal numbers as shown: i i i i + 2Jn−2 + Jn−3 + · · · + Jn−k , Jni = Jn−1

(1)

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Jacobsthal numbers

for n > 0 and 1 ≤ i ≤ k, with initial conditions

Jni

=

1 if i + n = 1, for 1 − k ≤ n ≤ 0, 0 otherwise,

where Jni is the nth term of the ith sequence. For k = 2 and i = 1 the generalized order-k Jacobsthal sequence is reduced to the the conventional Jacobsthal sequence. By the definition of generalized Jacobsthal numbers, we can write following vector recurrence relation ⎡ ⎤ ⎤ ⎡ i Jni Jn+1 ⎢ Ji ⎥ ⎢ Ji ⎥ n ⎢ n−1 ⎥ ⎥ ⎢ ⎢ Ji ⎥ ⎥ ⎢ Ji (2) ⎢ n−1 ⎥ = C ⎢ n−2 ⎥ , ⎢ ⎥ ⎥ ⎢ . . . . ⎣ ⎦ ⎦ ⎣ . . i Jn−k+2

i Jn−k+1

where C is a k-square companion ⎡ 1 ⎢ ⎢ 1 ⎢ ⎢ 0 C=⎢ ⎢ 0 ⎢ ⎢ .. ⎣ .

matrix as following: ⎤ 2 1 ··· 1 1 ⎥ 0 0 ··· 0 0 ⎥ ⎥ 1 0 ··· 0 0 ⎥ ⎥. 0 1 ··· 0 0 ⎥ ⎥ .. .. . . .. .. ⎥ . . . ⎦ . . 0 0 0 ··· 1 0

(3)

The matrix C is called to be generalized order-k Jacobsthal matrix. Let us to define a k−square matrix Bn = [bij ] to deal with the k sequences of the generalized order-k Jacobsthal numbers, as following: ⎡ ⎢ ⎢ Bn = ⎢ ⎣

Jn1 1 Jn−1 .. .

Jn2 2 Jn−1 .. .

··· ··· .. .

Jnk k Jn−1 .. .

⎤ ⎥ ⎥ ⎥ ⎦

(4)

1 2 k Jn−k+1 Jn−k+1 · · · Jn−k+1

If we expand (2) to k columns, we obtain the following matrix equation: Bn = CBn−1 . Then we have the following lemma.

(5)

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Lemma 1 Let C and Bn be as in (3) and (4), respectively. Then for all integers n ≥ 0 Bn = C n . Proof. By (4), we have Bn = CBn−1 . Then, by an inductive argument, we can write it as Bn = C n−1 B1 . By definition of the generalized order-k Jacobsthal numbers, B1 = C; therefore Bn = C n , which is desired. Corollary 2 Let Bn be as in (4). Then ⎧ ⎨ −2, if k = 2 det Bn = . 1, if k is odd ⎩ −1, if k is even, (k = 2) Proof. From Lemma 1, we know Bn = C n . Then det Bn = det C n = det(C)n . By the Laplace expansion of determinant with respect to the any column, it is easy to compute the determinant of C. So the proof is complete. Now we give some relations involving the generalized order-k Jacobsthal numbers. Lemma 3 Let Jni be the nth generalized order-k Jacobsthal number. Then 1 Jn+1 = Jn1 + Jn2 2 Jn+1 = 2Jn1 + Jn3 i Jn+1 = Jn1 + Jni+1 ; 3 ≤ i ≤ k − 1 k . Jn1 = Jn+1

Proof. We know from (4), Bn+1 = Bn B1 . By using the matrix multiplication the proof is readily seen. Some generalized order-k Jacobsthal numbers are given in Table 1.

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Jacobsthal numbers

n\i -2 -1 0 1 2 3 4 5

3

k= 1 0 0 1 1 3 5 11 21

2 2 0 1 0 2 2 6 10 22

k 1 0 0 1 1 3 6 13 28

= 3 2 3 0 1 1 0 0 0 2 1 3 1 7 3 15 6 32 13 Table1

1 0 0 1 1 3 6 14 30

k 2 0 1 0 2 3 8 16 37

= 3 1 0 0 1 2 4 9 20

4 4 0 0 0 1 1 3 6 14

Generalized Binet Formula

In this section we derived a generalized Binet formula for generalized order-k Jacobsthal numbers. From companion matrices, it is known that the chacteristic equation of the matrix C is xk − xk−1 − 2xk−2 − xk−3 − · · ·− x − 1 = 0, which is also the characteristic equation of generalized order-k Jacobsthal numbers. Lemma 4 The equation xk+1 − 2xk − xk−1 + xk−2 + 1 does not have multiple roots for k ≥ 3. Proof. Let f (x) = xk − xk−1 − 2xk−2 − xk−3 − · · · − x − 1 and g(x) = (x − 1)f (x) = xk+1 − 2xk − xk−1 + xk−2 + 1. It is easy to see that, 1 is a root of g(x) but not a multiple root, since k ≥ 3 and f (1) = 0. Suppose that β is a multiple root of g(x) such that β = 0 and β = 1. Since β is a multiple root g(β) = β k+1 − 2β k − β k−1 + β k−2 + 1 = β k−2[β 3 − 2β 2 − β + 1] + 1 = 0 and g (β) = (k + 1)β k − 2kβ k−1 − (k − 1)β k−2 + (k − 2)β k−3 = 0 = β k−3[(k + 1)β 3 − 2kβ 2 − (k − 1)β + k + 2] = 0.

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Thus

√ √ [a + 12 bk + b]1/3 1 14k 2 − 6 √ √ β1 = + 2k and + 3(k + 1) 2 [a + 12 bk + b]1/3

β2,3

1 = − 3(k + 1)

√ √ [a + 12 bk + b]1/3 7k 2 − 3 √ √ + 2k − 4 [a + 12 bk + b]1/3 √ √ √ 14k 2 − 6 3 [a + 12 bk + b]1/3 √ √ i − ± , 2 2 [a + 12 bk + b]1/3

where a = 28k 3 − 612k − 432k 2 − 216 and b = −147k 4 + 126k 3 + 1149k 2 + 1164k + 336. It is easy to see that βi ’s are distinct from each other. Hence 0 = −g(βi ) = βik−2[−βi3 + 2βi2 + βi − 1] − 1

(6)

= uk,i − 1, where uk,i = βik−2[−βi3 + 2βi2 + βi − 1]. By choosing k = 3 and 1 ≤ i ≤ 3, (6) can be written as 0 = −g(β1 ) = β1 [−β13 + 2β12 + β1 − 1] − 1 = u3,1 − 1 = 0. and u3,1 = −0.8445476618 = 1 is a contradiction. Similarly for β2 0 = −g(β2 ) = β2 [−β23 + 2β22 + β2 − 1] − 1 and u3,2 = β2 [−β23 + 2β22 + β2 − 1] = 1.172273831 + 0.3253556687i = 1 is a contradiction and by similar way for β3 , u3,3 = β3 [−β33 + 2β32 + β3 − 1] = 1.172273831 − 0.3253556687i = 1. This is also a contradiction because we did suppose that β is a multiple root for any integers k ≥ 3. Therefore, the equation g(x) = 0 does not have multiple roots. Consequently, from Lemma 4, the equation xk − xk−1 − 2xk−2 − xk−3 − · · · − x − 1 = 0 does not have multiple roots for k ≥ 3. Let f (λ) be the characteristic polynomial of the generalized order-k Jacobsthal matrix C. Then, by Lemma 4, λ1 , λ2 , . . . , λk are the distinct eigenvalues of matrix C. Let V be Vandermonde matrix as follows: ⎡ k−1 k−2 ⎤ λ1 λ1 · · · λ1 1 ⎢ λk−1 λk−2 · · · λ2 1 ⎥ 2 ⎢ 2 ⎥ V =⎢ . .. .. .. ⎥ . . . . ⎣ . . . . ⎦ . λk−2 · · · λk 1 λk−1 k k

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Jacobsthal numbers

Let cik be a k × 1 matrix

⎡ ⎢ ⎢ cik = ⎢ ⎣

⎤

λ1n+k−i λ2n+k−i .. .

⎥ ⎥ ⎥ ⎦

λkn+k−i (i)

and Vj be k−square matrix obtained from V by replacing the j th column of V by cik .Then we obtain the generalized Binet formula for the generalized order-k Jacobsthal numbers by the following theorem. Theorem 5 Let Jni be the nth term of ith Jacobsthal sequence, for 1 ≤ i ≤ k. Then j Jn−i+1 =

(i)

det(Vj ) . det(V )

Proof. C is diagonizable, due to its eigenvalues are distinct. Denote V T = D and D is invertible. Then we can write D −1 CD = diag(λ1 , λ2 , . . . , λk ) = Λ Hence C is similar to Λ. So we obtain C n D = DΛn . It is known that Bn = C n from Lemma 1. Then we have the following linear system of equations: + bi2 λk−2 + · · · + bik = λn+k−i bi1 λk−1 1 1 1 bi1 λk−1 + bi2 λk−2 + · · · + bik = λn+k−i 2 2 2 .. . bi1 λk−1 + bi2 λk−2 + · · · + bik = λn+k−i , k k k where Bn = [bij ]k×k . Thus we obtain for each j = 1, 2, ..., k, (i)

det(Vj ) . bij = det(V ) j i.e., bij = Jn−i+1 . So we have the conclusion.

4

Sums of the Jacobsthal numbers

In this section, we extend the matrix representation and obtain the sums of the generalized Jacobsthal numbers. The sums Tn of the generalized order-k Jacobsthal numbers are defined by Tn =

n−1 i=0

Ji1 .

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k Since Jn1 = Jn+1 which is given in Lemma 4, we can rewrite it as

Tn =

n

Jik .

i=1

Let E and Wn be (k + 1)-square matrices such that ⎡ ⎢ ⎢ ⎢ E=⎢ ⎢ ⎣

1 1 0 .. .

0 0 ··· 0 C

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

0 and ⎡ ⎢ ⎢ ⎢ Wn = ⎢ ⎢ ⎣

1 Tn Tn−1 .. .

0 0 ··· 0 Bn

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦

Tn−k+1 where C and Bn are k-square matrices as in (3) and (4), respectively. Using k the formula Jn1 = Jn+1 , we obtain 1 + Tn−1 Tn = Jn−1

so we derive the following matrix recurrence equation: Wn = Wn−1 E. By more generalization, we also have Wn = W1 E n−1 . Since T−i = 0; 1 ≤ i ≤ k and by the definition of the generalized orderk Jacobsthal numbers, we get W1 = E, and in general, Wn = E n . So we obtain the generating matrix for the sums of the generalized order-k Jacobsthal numbers. Since Wn = E n , we may write Wn+1 = Wn W1 = W1 Wn

(7)

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Jacobsthal numbers

which shows that W1 is commutative with Wn as well matrix multiplication. By an application of equation (7), the sums of the generalized order-k Jacobsthal numbers satisfy the recurrence relation : Tn = 1 + Tn−1 + 2Tn−2 +

k

Tn−i .

(8)

i=3

For example, when k = 2 and i = 1 the sequence is reduced to the usual Jacobsthal sequence. So the equation (8) becomes n−1

Tn =

Ji1 = 1 + Tn−1 + 2Tn−2 .

i=0 1 Since Tn = Jn−1 + Tn−1 , the sums of the Jacobsthal numbers are n−1 i=0

Ji =

3Jn−1 + 2Jn−2 − 1 . 2

Corollary 6 Let Ji be the ith Jacobsthal number. Then, n−1 i=0

⎧ ⎨ Ji =

⎩

3Jn−1 +2Jn−2 −1 2

= Jn , if n is even .

3Jn−1 +2Jn−2 −1 2

= Jn + 1, if n is odd

References [1] D. Kalman, Generalized Fibonacci numbers by matrix methods. Fibonacci Quart. 20 (1), (1982) 73-76. [2] E. Kılı¸c and D. Ta¸sçı, The Generalized Binet Formula, Representation and Sums of the Generalized order-k Pell numbers, Taiwanese Journal of Mathematics, 10, (6), (2006), 1661-1670. [3] D. D. Frey and J. A. Sellers, Jacobsthal Numbers and Alternating Sign Matrices, Journal of Integer Sequences, 3 (2000), Article 00.2.3. [4] F. Koken, D. Bozkurt, On the Jacobsthal numbers by matrix methods, Int. J. Contemp. Math. Sciences, 3, (13), (2008), 605-614. [5] D. Tasci and E. Kilic, On the Order-k Generalized Lucas Numbers, Appl.Math.Comput., 155, (3), (2004), 637-641.

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[6] E. Kilic and D. Tasci, On the Generalized Order-k Fibonacci and Lucas Numbers, Rocky Mountain J. Math., 36, 6, (2006),1915-1926. [7] E. Kilic, The generalized order-k Fibonacci–Pell sequence by matrix methods, Journal of Computational and Applied Mathematics 209, (2007) 133 – 145. Received: April, 2009