On the Smarandache-Pascal derived sequences of generalized Tribonacci numbers

Wu et al. Advances in Difference Equations 2013, 2013:284 http://www.advancesindifferenceequations.com/content/2013/1/284

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On the Smarandache-Pascal derived sequences of generalized Tribonacci numbers Zhengang Wu1* , Jianghua Li2 and Han Zhang1 * Correspondence: sky.wzgfff@163.com 1 Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R. China Full list of author information is available at the end of the article

Abstract For any sequence recurrence sequence {Tn } the Smarandache-Pascal derived

formula, n! denotes the of {bn } is defined by Tn+1 = nk=0 nk · bk+1 for all n ≥ 2, where nk = k!(n–k)! combination number. The recurrence formula of {Tn } is obtained by the properties of the third-order linear recurrence sequence.

Keywords: Smarandache-Pascal derived sequence; Tribonacci numbers; combination number; elementary method

1 Introduction For any sequence {bn }, a new sequence {Tn } is defined by the following method: T = b ,

T = b + b , T = b + b + b , generally, Tn+ = nk= nk · bk+ for all n ≥ , where n n! is the combination number. This sequence is called the Smarandache-Pascal = k!(n–k)! k derived sequence of {bn }. It was introduced by professor Smarandache in [] and studied by some authors. For example, Murthy and Ashbacher [] proposed a series of conjectures related to Fibonacci numbers and the Smarandache-Pascal derived sequence, one of them is as follows.

Conjecture Let {bn } = {Fn+ } = {F , F , F , F , . . .}, {Tn } be the Smarandache-Pascal derived sequence of {bn }, then we have the recurrence formula Tn+ =  · (Tn – Tn– ),

n ≥ .

Li and Han [] studied these problems and proved a generalized conclusion as follows.

Proposition Let {Xn } be a second-order linear recurrence sequence with X = u, X = v, Xn+ = aXn + bXn– for all n ≥ , where a + b > . For any positive integer d ≥ , we define the Smarandache-Pascal derived sequence of {Xdn+ } as Tn+ =

n

n k=

k

· Xdk+ .

Then we have the recurrence formula

Tn+ = ( + Ad + b · Ad– ) · Tn –  + Ad + b · Ad– + (–b)d · Tn– , where the sequence {An } is defined as A = , A = a, An+ = a · An + b · An– for all n ≥ . ©2013 Wu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Wu et al. Advances in Difference Equations 2013, 2013:284 http://www.advancesindifferenceequations.com/content/2013/1/284

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It is clear that if we take b = , then Xn is the Fibonacci polynomials, see [–].

The main purpose of this paper is, using the elementary method and the properties of the third-order linear recurrence sequence, to unify the above results by proving the following theorem.

Theorem Let {Xn } be a third-order linear recurrence sequence Xn+ = a · Xn+ + b · Xn+ + c · Xn with the initial values X = u, X = v and X = w for all n ≥ , where a, b and c are positive integers. For any positive integer d ≥ , we define the Smarandache-Pascal derived sequence of {Xdn+ } as Tn+ =

n

n k=

k

· Xdk+ .

Then we have the recurrence formula Tn+ =

g g – g g + g g g – g g – g g g g – g g · Tn + · Tn– + · Tn– , g – g g – g g – g

where g = f + f + c · Ad f ,

g = f Ad+ – f f + c · Ad f ,

g = c · Ad+ – c · Ad f + c · Ad f , g = c · Ad f ,

g = f Ad+ – f f + c · Ad (f + f ),

g = c · (Ad+ – Ad f + Ad f – Ad ),

g = c · Ad f

and f = b · Ad + c · Ad– + , f = b · Ad– + c · Ad , f =

Ad , Ad+

f =  + Ad+ , f =  + c · Ad– –

f = b · Ad– + c · Ad– –

c · Ad , Ad+

b · Ad + c · Ad– Ad + Ad , Ad+

the sequence {An } is defined by An+ = a · An+ + b · An+ + c · An with the initial values A = , A =  and A = a for all n ≥ .

From our theorem we know that if {bn } is a third-order linear recurrence sequence, then its Smarandache-Pascal derived sequence {Tn } is also a third-order linear recurrence sequence.

2 Proof of the theorem To complete the proof of our theorem, we need the following lemma.

Lemma Let integers m ≥  and n ≥ . If the sequence {Xn } satisfies the recurrence relations Xn+ = a · Xn+ + b · Xn+ + c · Xn , n ≥ , then we have the identity Xm+n = An · Xm+ + (b · An– + c · An– ) · Xm+ + c · An– · Xm ,

Wu et al. Advances in Difference Equations 2013, 2013:284 http://www.advancesindifferenceequations.com/content/2013/1/284

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where An is defined by An+ = a · An+ + b · An+ + c · An with the initial values A = , A =  and A = a for all n ≥ .

Proof Now we prove this lemma by mathematical induction. Note that the recurrence formula Xm+ = a · Xm+ + b · Xm+ + c · Xm = A · Xm+ + (b · A + c · A ) · Xm+ + c · A · Xm for all n ≥ . That is, the lemma holds for n =  since Xm+ = a · (a · Xm+ + b · Xm+ + c · Xm ) + b · Xm+ + c · Xm+

= a + b · Xm+ + (ab + c) · Xm+ + ac · Xm = A · Xm+ + (b · A + c · A ) · Xm+ + c · A · Xm .

That is, the lemma holds for n = . Suppose that for all integers  ≤ n ≤ k, we have Xm+n = An · Xm+ + (b · An– + c · An– ) · Xm+ + c · An– · Xm . Then, for n = k + , from the recurrence relations for Xm and the inductive hypothesis, we have Xm+k+ = a · Xm+k + b · Xm+k– + c · Xm+k–

= a · Ak · Xm+ + (b · Ak– + c · Ak– ) · Xm+ + c · Ak– · Xm

+ b · Ak– · Xm+ + (b · Ak– + c · Ak– ) · Xm+ + c · Ak– · Xm

+ c · Ak– · Xm+ + (b · Ak– + c · Ak– ) · Xm+ + c · Ak– · Xm

= (a · Ak + b · Ak– + c · Ak– ) · Xm+ + ab · Ak– + ac + b · Ak–

+ bc · Ak– + c · Ak– · Xm+ + c(a · Ak– + b · Ak– + c · Ak– ) · Xm

= Ak+ · Xm+ + ab · Ak– + b · Ak– + bc · Ak– + c · Ak– · Xm+ + c · Ak · Xm = Ak+ · Xm+ + (b · Ak + c · Ak– ) · Xm+ + c · Ak · Xm .

That is, the lemma also holds for n = k + . This completes the proof of our lemma by mathematical induction.

Now we use this lemma to complete the proof of our theorem. From the properties of

the binomial coefficient nk , we have

(n – )! n– n– (n – )! = + + k– k k!(n –  – k)! (k – )!(n – k)!

 n  (n – )! . = + = k (k – )!(n – k – )! k n – k

()

For any positive integer d, from the lemma we have Xdk+d+ = Ad+ · Xdk+ + (b · Ad + c · Ad– ) · Xdk+ + c · Ad · Xdk . By the definition of Tn , we may deduce that Tn+ =

n

n k=

k

· Xdk+

= X + Xdn+ +

n–

n– n– · Xdk+ + k– k k=

Wu et al. Advances in Difference Equations 2013, 2013:284 http://www.advancesindifferenceequations.com/content/2013/1/284

=

n–

n–

= Tn +

· Xdk+ +

k

k=

n–

n– k

k=

n–

n– k

k=

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· Xdk+d+ + Xdn+

· Xdk+d+

n–

n– · Ad+ · Xdk+ + (b · Ad + c · Ad– ) · Xdk+ + c · Ad · Xdk = Tn + k k=

= (b · Ad + c · Ad– + ) · Tn + Ad+

n–

n– k=

+ c · Ad

n–

n– k

k=

k

· Xdk+

· Xdk .

For convenience, we let f (Ak ) = b · Ad + c · Ad– +  (briefly f ), then the above identity implies that Tn+ = f · Tn + Ad+

n–

n– k

k=

· Xdk+ + c · Ad

n–

n– k

k=

· Xdk .

()

From this identity, we can also deduce Tn = f · Tn– + Ad+

n–

n– k

k=

· Xdk+ + c · Ad

n–

n– k

k=

· Xdk

and Tn– = f · Tn– + Ad+

n–

n– k

k=

· Xdk+ + c · Ad

n–

n– k=

k

· Xdk .

They are equivalent to

n–

n– k=

k

· Xdk+ =



Tn – f · Tn– – c · Ad

Ad+

n–

n– k=

k

· Xdk

()

and

n–

n– k=

k

· Xdk+ =



Tn– – f · Tn– – c · Ad

Ad+

n–

n– k=

k

· Xdk .

()

On the other hand, from the lemma we also deduce Xdk+d+ = Ad+ · Xdk+ + (b · Ad+ + c · Ad ) · Xdk+ + c · Ad+ · Xdk . Then we have

n–

n– k=

k

· Xdk+

= X + Xdn–d+ +

n–

n– k=

k

· Xdk+

Wu et al. Advances in Difference Equations 2013, 2013:284 http://www.advancesindifferenceequations.com/content/2013/1/284

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n–

n– n– · Xdk+ + = X + Xdn–d+ + k– k k=

=

n–

k=

=

k=

n–

k=

+

n–

n– n– · Xdk+d+ + · Xdk+ k k

n– · Ad+ · Xdk+ + (b · Ad+ + c · Ad ) · Xdk+ + c · Ad+ · Xdk k

n–

n– k=

k

= ( + Ad+ ) ·

· Xdk+

n–

n– k

k=

· Xdk+ + c · Ad+

n–

n– k

k=

· Xdk

+ (b · Ad+ + c · Ad ) · Tn– .

()

Similarly, applying formula () and identity (), we have

n–

n– k

k=

· Xdk

= X + Xdn–d +

n–

n– k

k=

= X + Xdn–d +

· Xdk

n–

n– n– · Xdk + k– k k=

=

n–

k=

+

n– · Ad · Xdk+ + (b · Ad– + c · Ad– ) · Xdk+ + c · Ad– · Xdk k

n–

n– k=

= Ad ·

k

· Xdk

n–

n– k=

k

· Xdk+ + ( + c · Ad– )

n–

n– k=

k

· Xdk

+ (b · Ad– + c · Ad– ) · Tn–

n–

Ad n– · Xdk Tn – (b · Ad + c · Ad– + ) · Tn– – c · Ad = k Ad+ k=

+ ( + c · Ad– )

n–

k=

n– · Xdk k

+ (b · Ad– + c · Ad– ) · Tn–

b · Ad + c · Ad– Ad + Ad Ad = · Tn + b · Ad– + c · Ad– – · Tn– Ad+ Ad+

n– c · Ad n –  · Xdk . +  + c · Ad– – k Ad+ k=

()

Wu et al. Advances in Difference Equations 2013, 2013:284 http://www.advancesindifferenceequations.com/content/2013/1/284

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For convenience, we let f = b · Ad+ + c · Ad ,

f =  + Ad+ , f =

Ad , Ad+

f =  + c · Ad– –

f = b · Ad– + c · Ad– –

c · Ad , Ad+

b · Ad + c · Ad– Ad + Ad , Ad+

then identities () and () imply that

n–

n– k

k=

= f ·

· Xdk+

n–

n– k

k=

n–

n– k

k=

· Xdk+ + c · Ad+

n–

n– k

k=

· Xdk = f

n–

n– k

k=

· Xdk + f · Tn– ,

()

· Xdk + f Tn + f Tn– .

()

Combining (), (), () and (), we deduce Tn+ = (f + f + c · Ad f ) · Tn + (f Ad+ – f f + c · Ad f ) · Tn–

n–

n– · Xdk . + c · Ad+ – c · Ad f + c · Ad f · k

()

k=

Applying formula (), we deduce

n– n– n– n– n– n . + + = + = k– k– k k– k k

()

From this and identities () and (), note that Xdk+d = Ad · Xdk+ + (b · Ad– + c · Ad– ) · Xdk+ + c · Ad– · Xdk , we have

n–

n– k

k=

· Xdk

n–

n– n– · Xdn–d + · Xdk n– k

= X + Xdn–d +

k=

= X + Xdn–d + (n – ) · Xdn–d +

n–

n– n– n– · Xdk + + k– k– k k=

= X + Xdn–d + (n – ) · Xdn–d +

n–

k=

+

n–

k=

=

n–

k=

n– · Xdk – X k

n–

n– n– · Xdk+d – Xdn–d + · Xdk+d – (n – ) · Xdn–d – Xdn–d k k k=

n– n–

n– n– n– · Xdk + · Xdk+d + · Xdk+d k k k k=

k=

Wu et al. Advances in Difference Equations 2013, 2013:284 http://www.advancesindifferenceequations.com/content/2013/1/284

= Ad

n–

n– k=

k

n–

n–

· Xdk+ + Ad

k

k=

+ (c · Ad– + )

n–

n– k

k=

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· Xdk+ + c · Ad–

n–

n– k=

k

· Xdk

· Xdk + (b · Ad– + c · Ad– ) · (Tn– + Tn– )

b · Ad + c · Ad– Ad Ad = Tn + b · Ad– + c · Ad– – · Tn– Ad+ Ad+

b · Ad + c · Ad– Ad + Ad + b · Ad– + c · Ad– – · Tn– Ad+

n– c · Ad n –  · Xdk + c · Ad– – k Ad+ k=

c · Ad + c · Ad– +  – Ad+

n– k=

n– · Xdk k

= f Tn + (f + f )Tn– + f Tn– + (f – )

n–

n– k

k=

· Xdk + f

n–

n– k=

k

· Xdk .

()

Combining (), (), () and (), we deduce

Tn+ = (f + f + c · Ad f ) · Tn + f Ad+ – f f + c · Ad (f + f ) · Tn– + c · Ad f · Tn–

n–

n– · Xdk + c(Ad+ – Ad f + Ad f – Ad ) · k k=

+ c · Ad f ·

n–

k=

n– · Xdk . k

()

From identity () we can also deduce Tn = (f + f + c · Ad f ) · Tn– + (f Ad+ – f f + c · Ad f ) · Tn–

n–

n–  · Xdk . + c · Ad+ – c · Ad f + c · Ad f · k

()

k=

For convenience, we let g = f + f + c · Ad f ,

g = f Ad+ – f f + c · Ad f ,

g = c · Ad+ – c · Ad f + c · Ad f , g = c · Ad f ,

g = f Ad+ – f f + c · Ad (f + f ),

g = c · (Ad+ – Ad f + Ad f – Ad ),

g = c · Ad f .

Inserting () and () into (), we deduce Tn+ =

g g – g g + g g g – g g – g g g g – g g · Tn + · Tn– + · Tn– . g – g g – g g – g

This completes the proof of our theorem.

()

Wu et al. Advances in Difference Equations 2013, 2013:284 http://www.advancesindifferenceequations.com/content/2013/1/284

Remark In fact, using the above formulas, we can also obtain the recurrence formula of the Smarandache-Pascal derived sequence {Tn } of {un }, where {un } denotes the mth-order linear recursive sequences as follows: un = a un– + a un– + · · · + am– un–m+ + am un–m , with initial values ui ∈ N for n > m and  ≤ i < m.

Competing interests The authors declare that they have no competing interests.

Authors’ contributions ZW obtained the theorems and completed the proof. JL and HZ corrected and improved the final version. All authors read and approved the final manuscript.

Author details 1 Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R. China. 2 College of Science, Xi’an University of Technology, Xi’an, Shaanxi, P.R. China.

Acknowledgements The authors express their gratitude to the referee for very helpful and detailed comments. This work is supported by the N.S.F. (11071194, 11001218) of P.R. China and the G.I.C.F. (YZZ12062) of NWU. Received: 8 August 2013 Accepted: 29 August 2013 Published: 07 Nov 2013 References 1. Smarandache, F: Only Problems, Not Solutions. Xiquan Publishing House, Chicago (1993) 2. Murthy, A, Ashbacher, C: Generalized Partitions and New Ideas on Number Theory and Smarandache Sequences.

Hexis, Phoenix (2005) 3. Li, X, Han, D: On the Smarandache-Pascal derived sequences and some of their conjectures. Adv. Differ. Equ. 2013, 240 (2013) 4. Ma, R, Zhang, W: Several identities involving the Fibonacci numbers and Lucas numbers. Fibonacci Q. 45, 164-170 (2007) 5. Yi, Y, Zhang, W: Some identities involving the Fibonacci polynomials. Fibonacci Q. 40, 314-318 (2002) 6. Wang, T, Zhang, W: Some identities involving Fibonacci, Lucas polynomials and their applications. Bull. Math. Soc. Sci. Math. Roum. 55, 95-103 (2012) 7. Riordan, J: Combinatorial Identities. Wiley, New York (1968)

10.1186/1687-1847-2013-284 Cite this article as: Wu et al.: On the Smarandache-Pascal derived sequences of generalized Tribonacci numbers.

Advances in Difference Equations 2013, 2013:284

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