THE GENERALIZED BINET FORMULA, REPRESENTATION AND SUMS OF THE GENERALIZED ORDER-$k$ PELL NUMBERS

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

“…Recall that the (k + 1) × (k + 1) matrix A be as in (13). We consider the characteristic equation of matrix A, then the characteristic polynomial of A is…”

“…. , k ) and the matrices A and G given by (13) and (19), respectively. If we compute the det G by the Laplace expansion according to the first row, then, by the definition of k × k Vandermonde matrix V , we easily obtain det G = det V .…”

“…Also in [12], the authors give the relationships between the generalized order-k Fibonacci numbers g i n and the generalized order-k Lucas numbers (see for more details about the generalized Lucas numbers [23]), and give the some useful identities and Binet formulas of the generalized order-k Fibonacci sequence {g i n } and Lucas sequence {l i n }. In [13], the authors define the k sequences of the generalized order-k Pell numbers as follows: for n > 0 and 1 i k P i n = 2P i n−1 + P i n−2 + · · · + P i n−k ,…”

“…Further, the combinatorial matrix theory is very important tool to obtain results for number theory [2]. In [15,12,13], the authors define certain generalizations of the usual Fibonacci, Pell and Lucas numbers by matrix methods and then obtain the Binet formulas and combinatorial representations of the generalizations of these number sequence. Furthermore, using matrix methods for computing of properties of recurrence relations are very convenient to parallel algorithm in computer science (see [4,6,7,18,21,22,25]).…”

Sums of the squares of terms of sequence {u n }

“…Some authors derive the generating matrices for certain linear recurrences (for more details, see [1], [6][7][8], [11], [19], [21], [26], [29]). For example, the terms of generalized Fibonacci sequence {u n } can be generated by the powers of the following 2 × 2 companion matrix:…”