The 99th Fibonacci identity.
The (ABC) Conjecture and the radical index of integers
The Bernoullian of a Matrix. (A Generalization of the Bernoulli Numbers)
Si associano ad una matrice infinita di un certo tipo altre due matrici dello stesso tipo, dette rispettivamente bernoulliana e antibernoulliana di A. Si studiano alcune proprietà di queste matrici. Si ottiene in tal via una generalizzazione dei classici numeri di Bernoulli.
The combinatorics of certain -ary meta-Fibonacci sequences.
The diophantine equations .
The distribution of Farey numbers.
The Distribution of Farey Points.
The distribution of second order linear recurrence sequences mod
The Family of Metallic Means
The Fibonacci number of a grid graph and a new class of integer sequences.
The Golden Section, Fibonacci Series and New Hyperbolic Models of Nature
The k-Fibonacci matrix and the Pascal matrix
We define the k-Fibonacci matrix as an extension of the classical Fibonacci matrix and relationed with the k-Fibonacci numbers. Then we give two factorizations of the Pascal matrix involving the k-Fibonacci matrix and two new matrices, L and R. As a consequence we find some combinatorial formulas involving the k-Fibonacci numbers.
The Pfaffian transform.
The power of a prime that divides a generalized binomial coefficient.
The power of powerful numbers.
The terms in Lucas sequences divisible by their indices.
The terms of the form 7kx² in the generalized Lucas sequence with parameters P and Q
Let Vₙ(P,Q) denote the generalized Lucas sequence with parameters P and Q. For all odd relatively prime values of P and Q such that P² + 4Q > 0, we determine all indices n such that Vₙ(P,Q) = 7kx² when k|P. As an application, we determine all indices n such that the equation Vₙ = 21x² has solutions.
The Tribonacci substitution.
Tiling a -board with squares and dominoes.
Tiling proofs of some Fibonacci-Lucas relations.