Abstract:
In the paper, defining the Generalized Fibonacci Sequences * The general formula of Generalized Fibonacci Sequences * is given by using characteristic equation. Using the recurrence property of generalized Fibonacci Sequences, this article provides the some finite sum formulas which are * for generalized Fibonacci Sequences, promoting the conclusion of Generalized Fibonacci Seqquences.(* Indicates a formula, please see the full text)

Abstract:
In this paper, we define a new family of k-generalized Fibonacci numbers. Furthermore, we give sums and recurrence relations of this numbers and obtain generating functions of this numbers for k=2.

Abstract:
The scope of the paper is the definition and discussion of the polynomial generalizations of the {sc Fibonacci} numbers called here $delta$-{sc Fibonacci} numbers. Many special identities and interesting relations for these new numbers are presented. Also, different connections between $delta$-{sc Fibonacci} numbers and {sc Fibonacci} and {sc Lucas} numbersare proven in this paper.

Abstract:
In this paper, we study the k–Lucas numbers of arithmetic indexes of the form an+r , where n is a natural number and r is less than r. We prove a formula for the sum of these numbers and particularly the sums of the first k-Lucas numbers, and then for the even and the odd k-Lucas numbers. Later, we find the generating function of these numbers. Below we prove these same formulas for the alternated k-Lucas numbers. Then, we prove a relation between the k–Fibonacci numbers of indexes of the form 2rn and the k–Lucas numbers of indexes multiple of 4. Finally, we find a formula for the sum of the square of the k-Fibonacci even numbers by mean of the k–Lucas numbers.

Abstract:
In this paper we present some identities for the sums of squares of Fibonacci and Lucas numbers with consecutive primes, using maximal prime gap G(x)~log^{2}x, as indices.

Abstract:
When we study a congruence T(x) ≡ ax modulo m as pseudo random number generator, there are several means of ensuring the independence of two successive numbers. In this report, we show that the dependence depends on the continued fraction expansion of m/a. We deduce that the congruences such that m and a are two successive elements of Fibonacci sequences are those having the weakest dependence. We will use this result to obtain truly random number sequences x_{n}. For that purpose, we will use non-deterministic sequences y_{n}. They are transformed using Fibonacci congruences and we will get by this way sequences x_{n}. These sequences x_{n} admit the IID model for correct model.

Abstract:
In this paper, we will see that
some k -Fibonacci sequences
are related to the classical Fibonacci sequence of such way that we can express the terms of a k -Fibonacci sequence in function of some terms of the
classical Fibonacci sequence. And the formulas will
apply to any sequence of a certain set of k' -Fibonacci sequences. Thus we find k -Fibonacci sequences relating to other k -Fibonacci sequences when σ'_{k} is linearly dependent of .

Abstract:
We study the eight infinite sequences of triples of natural numbers A=(F_{2n+1},4F_{2n+3},F_{2n+7}), B=(F_{2n+1},4F_{2n+5},F_{2n+7}), C=(F_{2n+1},5F_{2n+1},F_{2n+3}), D=(F_{2n+3},4F_{2n+1},F_{2n+3}) and A=(L_{2n+1},4L_{2n+3},L_{2n+7}), B=(L_{2n+1},4L_{2n+5},L_{2n+7}), C=(L_{2n+1},5L_{2n+1},L_{2n+3}), D=(L_{2n+3},4L_{2n+1},L_{2n+3}. The sequences A,B,C and D are built from the Fibonacci numbers F_{n} while the sequences A, B, C and D from the Lucas numbers L_{n}. Each triple in the sequences A,B,C and D has the property D(-4) (i. e., adding -4 to the product of any two different components of them is a square). Similarly, each triple in the sequences A, B, C and D has the property D(20). We show some interesting properties of these sequences that give various methods how to get squares from them.

Abstract:
Let us define A=H_{r}=(a_{ij})？to be n×n？r-Hankel
matrix. The entries of matrix A are F_{n}=F_{i+j-2}？or L_{n}=F_{i+j-2}？where F_{n}？and L_{n}？denote the usual Fibonacci and Lucas numbers,
respectively. Then, we
obtained upper and lower bounds for the spectral norm of matrix A. We compared our bounds with exact
value of matrix A’s spectral norm.
These kinds of matrices have