The generalized order-k Fibonacci–Pell sequence by matrix methods

Abstract

In this paper, we consider the usual and generalized order-k Fibonacci and Pell recurrences, then we define a new recurrence, which we call generalized order-k F-P sequence. Also we present a systematic investigation of the generalized order-k F-P sequence. We give the generalized Binet formula, some identities and an explicit formula for sums of the generalized order-k F-P sequence by matrix methods. Further, we give the generating function and combinatorial representations of these numbers. Also we present an algorithm for computing the sums of the generalized order-k Pell numbers, as well as the Pell numbers themselves.

On Distance(r,k)-Fibonacci Numbers and Their Combinatorial and Graph Interpretations

“…Assume that the condition (c3) is satisfied. Then the subfamily R (3) , we will call a decomposition with repetitions of the set with the rest at the end or at the beginning. Theorem 1.…”

“…For example, in [2] -Fibonacci numbers were introduced and defined recurrently for any integer ≥ 1 by ( , ) = ( , − 1) + ( , − 2) for ≥ 2 with ( , 0) = 0, ( , 1) = 1. In [3] the following generalization of the Fibonacci numbers was defined: = 2 −1 + −2 for an integer ≥ 0 such that 4 −1 + 1 ̸ = 0 and ≥ 2 with 0 = 0 and 1 = 1. Other interesting generalizations of Fibonacci numbers are presented in [4,5].…”

“…(5) Reference [9]. Consider (3) ( , ) = (3) ( , − + 1)+ (3) ( , − ) for ≥ 2 −1 with (3) ( , ) = 1 for = 0, . .…”

Tribonacci numbers with indices in arithmetic progression and their sums

“…The matrix method is very useful method in order to obtain some identities for special sequences. For example, using matrix methods, the authors obtained some identities for various special sequences (see [3,4,[6][7][8]) .…”

On generalizations of two curious divisibility properties

“…1/ .r 1/.kC1/ q r.k 1/ U k n r .p; q/ is divisible by U 2 n .p; q/: To do that we use matrix methods. Matrix methods are useful tools for derivating some properties of linear recurrences (see [3,5,6,8,10]). We consider the quotient…”