摘要：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 = ∑ k = 0 n ( 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 we have the recurrence formula T n + 1 = 49 ⋅ ( T n − T n − 1 ) , n ≥ 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.
关键词：Smarandache-Pascal derived sequence ; Fibonacci number ; combination number ; elementary method ; conjecture
