Abstract
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.
Highlights
For any sequence {bn}, we define a new sequence {Tn} through the following method:T = b, T = b + b, T = b + b + b, generally, Tn+ = n k= n k· bk+ for all n ≥, where n kThis sequence is called the Smarandache-Pascal derived sequence of {bn}
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 {bn} = {F1, F9, F17, . . .}, we have the recurrence formula Tn+1 = 49 · (Tn – Tn–1), n ≥ 2
Murthy and Ashbacher [ ] proposed a series of conjectures related to Fibonacci numbers and its Smarandache-Pascal derived sequence; three of them are as follows
Summary
For any sequence {bn}, the Smarandache-Pascal derived sequence {Tn} of {bn} is defined as T1 = b1, T2 = b1 + b2, T3 = b1 + 2b2 + b3, generally, Tn+1 = . .}, we have the recurrence formula Tn+1 = 49 · (Tn – Tn–1), n ≥ 2. Murthy and Ashbacher [ ] proposed a series of conjectures related to Fibonacci numbers and its Smarandache-Pascal derived sequence; three of them are as follows.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
