On the Generalized Order-$k$ Fibonacci and Lucas Numbers

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:…”

“…Also some authors have derived interesting relationships between the determinant or permanent of certain matrices and the linear recurrence relations (see [2][3][4], [9][10][11][12][13][14][15][16][17][18][19], [21][22][23][24][25][26][27][28]). These relationships are valid for both second order linear recurrences and higher order linear recurrences.…”

“…For example, some authors use matrix methods for derivation of sums, combinatorial representations and generalized Binet formula of the second order or higher order Fibonacci numbers and their certain generalizations [21], [5], [6], [19], [11]. In [11], the authors define a (k × k) matrix and use the nth power. Then they derive the generalized Binet formula, useful identities and combinatorial representations of the generalized order-k Fibonacci numbers.…”

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

“…Also Er give the generating matrix for the generalized order-k Fibonacci sequence {g i n }. 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]).…”