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.
Tiling proofs of some formulas for the Pell numbers of odd index.
Transforming recurrent sequences by using the binomial and invert operators.
Trees and meta-Fibonacci sequences.
Tribonacci modulo and
Our previous research was devoted to the problem of determining the primitive periods of the sequences where is a Tribonacci sequence defined by an arbitrary triple of integers. The solution to this problem was found for the case of powers of an arbitrary prime . In this paper, which could be seen as a completion of our preceding investigation, we find solution for the case of singular primes .
Tribonacci modulo
Our research was inspired by the relations between the primitive periods of sequences obtained by reducing Tribonacci sequence by a given prime modulus and by its powers , which were deduced by M. E. Waddill. In this paper we derive similar results for the case of a Tribonacci sequence that starts with an arbitrary triple of integers.
Trivial points on towers of curves
In order to study the behavior of the points in a tower of curves, we introduce and study trivial points on towers of curves, and we discuss their finiteness over number fields. We relate the problem of proving that the only rational points are the trivial ones at some level of the tower, to the unboundeness of the gonality of the curves in the tower, which we show under some hypothesis.
Two conjectures by Zhi-Hong Sun
Über den Rang gewisser zirkulanter Matrizen (zu Problem 764A).