Partial Sums for Second-Order Recurrence Sequences
Alwyn F. Horadam · The Fibonacci Quarterly · 1994
Motivation for this paper comes from a short article [4] in which some relations between a generalized Fibonacci sequence and the sequence of its partial sums were investigated. An opportunity was clearly provided for a deeper exploration of this theme. Accordingly, the purpose of this paper is (a) to extend the relations in [4] to generalized Pell numbers with (i) positive and (ii) negative subscripts, and (b) as an addendum, to expand the results in [4] to generalized Fibonacci numbers having negative subscripts. Consider the generalized Pell sequence {Pn} defined for all integers n by Pn+2=2Pn+l + P „ Pl = a,P2=b(P0=b-2a). (1.1) When a = 1, 6 = 2, the ordinary Pell sequence {p„} is generated, while when a = 1, b = 3, we derive the sequence {qn} defined by °n+2 = 2?»+l + In 1 \\ = 1, ft = 3 (*> =1) (1-2) so that q„=jQ„, the w * Pell-Lucas number [2]. Thus, we have the tabulation: n: Pn-Qn'-