Sentence examples for recurrence sequence from inspiring English sources

It is clear that (F_{n}) is a second-order linear recurrence sequence.

From our theorem we know that if { b n } is a third-order linear recurrence sequence, then its Smarandache-Pascal derived sequence { T n } is also a third-order linear recurrence sequence.

The recurrence formula of { T n } is obtained by the properties of the third-order linear recurrence sequence.

Let (S_{n}) be an rth-order linear recurrence sequence satisfying the recursion (S_{n+r}=sum_{k=0}^{r-1}c_{k}S_{n+k} ).

For any integer n ≥ 0, the well-known Pell numbers P n are defined by the second-order linear recurrence sequence P n + 2 = 2 P n + 1 + P n, where P 0 = 0 and P 1 = 1.

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.

We consider the following type of higher-order recurrence sequences.

Recently, many authors have studied some special linear recurrence sequences in algebraic structures; see, for example, [4 14].

Our results can be applied to any linear recurrence sequences by using a similar method; for example, Lucas numbers, Pell numbers, Horadam numbers, generalized Fibonacci p-numbers.

The recurrence sequences determined by equalities (5.1), (5.2) and (5.3), (5.4) arise in a natural way when boundary value problems of type (4.1), (4.1) and (4.3), (4.4) are considered.

Let α = 1 2 ( x + x 2 + 4 ) and β = 1 2 ( x − x 2 + 4 ), then from the properties of second-order linear recurrence sequences, we have F n ( x ) = α n − β n x 2 + 4 and L n ( x ) = α n + β n.