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

**Author Name:** Xiaoxue Li, Di Han

For any sequence $\{b_n\}$, the Smarandache-Pascal derived sequence $\{T_n\}$ of $\{b_n\}$ is defined as $\ T_1=b_1$, $T_2=b_1+b_2$, $T_3=b_1+2b_2+b_3$, generally, $T_{n+1}=\displaystyle \sum_{k=0}^{n}\binom{n}{k}\cdot b_{k+1}$ for all $n\geq 2$, where $\binom{n}{k}=\frac{n!}{k!(n-k)!}$ is the combination number. In reference [2], Amarnath Murthy and Charles Ashbacher proposed a series of conjectures related to Fibonacci numbers and its Smarandache-Pascal derived sequence, one of them is that if $\{b_n\}= \{F_1, F_9, F_{17}, \cdots\}$, then we have the recurrence formula $\ T_{n+1}=49\cdot \left(T_n-T_{n-1}\right)$, $n\geq 2$. 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 prove a generalized

Date of published: 2013-09-27

Journal Name: Advances In Difference Equations

DOI: Not Available

Keywords: Advances In Difference Equations

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