On the Smarandache-Pascal derived sequences and some of their conjectures
© Li and Han; licensee Springer 2013
Received: 9 June 2013
Accepted: 23 July 2013
Published: 8 August 2013
For any sequence , the Smarandache-Pascal derived sequence of is defined as , , , generally, for all , where 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 , then we have the recurrence formula , . 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.
For any sequence , we define a new sequence through the following method: , , , generally, for all , where is the combination number. This sequence is called the Smarandache-Pascal derived sequence of . 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 its Smarandache-Pascal derived sequence; three of them are as follows.
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.
Now we take , then from our theorem, we may immediately deduce the following three corollaries.
If we take , , in Corollary 1, then is a Fibonacci sequence. Note that , , , ; from Corollary 1, we may immediately deduce that the three conjectures above are true.
If we take , , and for all , then are the Pell numbers. From Corollary 1, we can also deduce the following.
On the other hand, from our theorem, we know that if is a second-order linear recurrence sequence, then its Smarandache-Pascal derived sequence is also a second-order linear recurrence sequence.
2 Proof of the theorem
To complete the proof of our theorem, we need the following.
That is, the lemma also holds for . This completes the proof of our lemma by mathematical induction. □
Now, our theorem follows from formula (6).
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.
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