# On the Smarandache-Pascal derived sequences and some of their conjectures

- Xiaoxue Li
^{1}and - Di Han
^{1}Email author

**2013**:240

https://doi.org/10.1186/1687-1847-2013-240

© Li and Han; licensee Springer 2013

**Received: **9 June 2013

**Accepted: **23 July 2013

**Published: **8 August 2013

## Abstract

For any sequence $\{{b}_{n}\}$, the Smarandache-Pascal derived sequence $\{{T}_{n}\}$ of $\{{b}_{n}\}$ is defined as ${T}_{1}={b}_{1}$, ${T}_{2}={b}_{1}+{b}_{2}$, ${T}_{3}={b}_{1}+2{b}_{2}+{b}_{3}$, generally, ${T}_{n+1}={\sum}_{k=0}^{n}\left(\genfrac{}{}{0ex}{}{n}{k}\right)\cdot {b}_{k+1}$ for all $n\ge 2$, where $\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}$ is the combination number. In reference (Murthy and Ashbacher in Generalized Partitions and New Ideas on Number Theory and Smarandache Sequences, 2005), authors proposed a series of conjectures related to Fibonacci numbers and its Smarandache-Pascal derived sequence, one of them is that if $\{{b}_{n}\}=\{{F}_{1},{F}_{9},{F}_{17},\dots \}$, then we have the recurrence formula ${T}_{n+1}=49\cdot ({T}_{n}-{T}_{n-1})$, $n\ge 2$. The main purpose of this paper is using the elementary method and the properties of the second-order linear recurrence sequence to study these problems and to prove a generalized conclusion.

## Keywords

## 1 Introduction

For any sequence $\{{b}_{n}\}$, we define a new sequence $\{{T}_{n}\}$ through the following method: ${T}_{1}={b}_{1}$, ${T}_{2}={b}_{1}+{b}_{2}$, ${T}_{3}={b}_{1}+2{b}_{2}+{b}_{3}$, generally, ${T}_{n+1}={\sum}_{k=0}^{n}\left(\genfrac{}{}{0ex}{}{n}{k}\right)\cdot {b}_{k+1}$ for all $n\ge 2$, where $\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}$ is the combination number. This sequence is called the Smarandache-Pascal derived sequence of $\{{b}_{n}\}$. It was introduced by professor Smarandache in [1] and studied by some authors. For example, Murthy and Ashbacher [2] proposed a series of conjectures related to Fibonacci numbers and its Smarandache-Pascal derived sequence; three of them are as follows.

**Conjecture 1**

*Let*$\{{b}_{n}\}=\{{F}_{8n+1}\}=\{{F}_{1},{F}_{9},{F}_{17},{F}_{25},\dots \}$, $\{{T}_{n}\}$

*be the Smarandache*-

*Pascal derived sequence of*$\{{b}_{n}\}$,

*then we have the recurrence formula*

**Conjecture 2**

*Let*$\{{b}_{n}\}=\{{F}_{10n+1}\}=\{{F}_{1},{F}_{11},{F}_{21},{F}_{31},\dots \}$, $\{{T}_{n}\}$

*be the Smarandache*-

*Pascal derived sequence of*$\{{b}_{n}\}$,

*then we have the recurrence formula*

**Conjecture 3**

*Let*$\{{b}_{n}\}=\{{F}_{12n+1}\}=\{{F}_{1},{F}_{13},{F}_{25},{F}_{37},\dots \}$, $\{{T}_{n}\}$

*be the Smarandache*-

*Pascal derived sequence of*$\{{b}_{n}\}$,

*then we have the recurrence formula*

Regarding these conjectures, it seems that no one has studied them yet; at least, we have not seen any related results before. These conjectures are interesting; they reveal the profound properties of the Fibonacci numbers. The main purpose of this paper is using the elementary method and the properties of the second-order linear recurrence sequence to study these problems and to prove a generalized conclusion. That is, we shall prove the following.

**Theorem**

*Let*$\{{X}_{n}\}$

*be a second*-

*order linear recurrence sequence with*${X}_{0}=u$, ${X}_{1}=v$, ${X}_{n+1}=a{X}_{n}+b{X}_{n-1}$

*for all*$n\ge 1$,

*where*${a}^{2}+4b>0$.

*For any positive integer*$d\ge 2$,

*we define the Smarandache*-

*Pascal derived sequence of*$\{{X}_{dn+1}\}$

*as*

*Then we have the recurrence formula*

*where the sequence*$\{{A}_{n}\}$

*is defined as*${A}_{0}=1$, ${A}_{1}=a$, ${A}_{n+1}=a\cdot {A}_{n}+b\cdot {A}_{n-1}$

*for all*$n\ge 1$.

*In fact this time*,

*the general term is*

Now we take $b=1$, then from our theorem, we may immediately deduce the following three corollaries.

**Corollary 1**

*Let*$\{{X}_{n}\}$

*be a second*-

*order linear recurrence sequence with*${X}_{0}=u$, ${X}_{1}=v$, ${X}_{n+1}=a{X}_{n}+{X}_{n-1}$

*for all*$n\ge 1$.

*For any even number*$d\ge 2$,

*we have the recurrence formula*

**Corollary 2**

*Let*$\{{X}_{n}\}$

*be a second*-

*order linear recurrence sequence with*${X}_{0}=u$, ${X}_{1}=v$, ${X}_{n+1}=a{X}_{n}+{X}_{n-1}$

*for all*$n\ge 1$.

*For any odd number*$d\ge 2$,

*we have the recurrence formula*

*where*

It is clear that ${F}_{n+1}(a)={A}_{n}(a)$ is a polynomial of *a*; sometimes, it is called a Fibonacci polynomial, because ${F}_{n}(1)={F}_{n}$ is Fibonacci number, see [3–5].

If we take $a=1$, ${X}_{0}=0$, ${X}_{1}=1$ in Corollary 1, then $\{{X}_{n}\}=\{{F}_{n}\}$ is a Fibonacci sequence. Note that ${A}_{n}={F}_{n+1}$, $2+{A}_{8}+{A}_{6}=2+{F}_{9}+{F}_{7}=2+34+13=49$, $2+{A}_{10}+{A}_{8}=2+{F}_{11}+{F}_{9}=2+89+34=125$, $2+{A}_{12}+{A}_{10}=2+{F}_{13}+{F}_{11}=2+233+89=324$; from Corollary 1, we may immediately deduce that the three conjectures above are true.

If we take $a=2$, ${X}_{0}={P}_{0}=0$, ${X}_{1}={P}_{1}=1$ and ${P}_{n+1}=2{P}_{n}+{P}_{n-1}$ for all $n\ge 1$, then ${P}_{n}$ are the Pell numbers. From Corollary 1, we can also deduce the following.

**Corollary 3**

*Let*${P}_{n}$

*be the Pell number*.

*Then for any positive integer*

*d*

*and*

*we have the recurrence formula*

On the other hand, from our theorem, we know that if $\{{b}_{n}\}$ is a second-order linear recurrence sequence, then its Smarandache-Pascal derived sequence $\{{T}_{n}\}$ is also a second-order linear recurrence sequence.

## 2 Proof of the theorem

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

**Lemma**

*Let integers*$m\ge 0$

*and*$n\ge 2$.

*If the sequence*$\{{X}_{n}\}$

*satisfying the recurrence relations*${X}_{n+2}=a\cdot {X}_{n+1}+b\cdot {X}_{n}$, $n\ge 0$,

*then we have the identity*

*where*${A}_{n}$

*is defined as*${A}_{0}=1$, ${A}_{1}=a$

*and*${A}_{n+1}=a\cdot {A}_{n}+b\cdot {A}_{n-1}$

*for all*$n\ge 1$,

*or*

*Proof*Now we prove this lemma by mathematical induction. Note that the recurrence formula ${X}_{m+2}=a\cdot {X}_{m+1}+b\cdot {X}_{m}$, ${A}_{1}=a$, ${A}_{0}=1$, ${A}_{n+1}=a\cdot {A}_{n}+b\cdot {A}_{n-1}$ for all $n\ge 1$. So ${X}_{m+2}={A}_{1}\cdot {X}_{m+1}+b\cdot {A}_{0}\cdot {X}_{m}$. That is, the lemma holds for $n=2$. Since ${X}_{m+3}=a\cdot {X}_{m+2}+b\cdot {X}_{m+1}=a\cdot (a\cdot {X}_{m+1}+b\cdot {X}_{m})+b\cdot {X}_{m+1}=({a}^{2}+b)\cdot {X}_{m+1}+ba\cdot {X}_{m}={A}_{2}\cdot {X}_{m+1}+b{A}_{1}\cdot {X}_{m}$. That is, the lemma holds for $n=3$. Suppose that for all integers $2\le n\le k$, we have ${X}_{m+n}={A}_{n-1}\cdot {X}_{m+1}+b\cdot {A}_{n-2}\cdot {X}_{m}$. Then for $n=k+1$, from the recurrence relations for ${X}_{m}$ and the inductive hypothesis, we have

That is, the lemma also holds for $n=k+1$. This completes the proof of our lemma by mathematical induction. □

*d*, from the definition of ${T}_{n}$ and the properties of the binomial coefficient $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$, we have

Now, our theorem follows from formula (6).

## Declarations

### Acknowledgements

The authors would like to thank the referee for carefully examining this paper and providing a number of important comments. This work is supported by the N.S.F. (11071194, 11001218) of P.R. China and G.I.C.F. (YZZ12062) of NWU.

## Authors’ Affiliations

## References

- Smarandache F:
*Only Problems, Not Solutions*. Xiquan Publishing House, Chicago; 1993.Google Scholar - Murthy A, Ashbacher C:
*Generalized Partitions and New Ideas on Number Theory and Smarandache Sequences*. Hexis, Phoenix; 2005:79.Google Scholar - Rong M, Wenpeng Z: Several identities involving the Fibonacci numbers and Lucas numbers.
*Fibonacci Q.*2007, 45: 164–170.Google Scholar - Yuan Y, Wenpeng Z: Some identities involving the Fibonacci polynomials.
*Fibonacci Q.*2002, 40: 314–318.Google Scholar - Tingting W, Wenpeng Z: Some identities involving Fibonacci, Lucas polynomials and their applications.
*Bull. Math. Soc. Sci. Math. Roum.*2012, 55(1):95–103.MathSciNetGoogle Scholar

## Copyright

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