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

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, Tn+1 = Sigma(n)(k=0)((n)(k)) . b(k+1) for all n >= 2, where ((n)(k)) = 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, ... }, then Tn+1 = 49 . (T-n - Tn-1), n >= 2 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.