Five new classes of Fibonacci-Hessenberg matrices are introduced. Further, we introduce the notion of two-dimensional Fibonacci arrays and show that three classes of previously known Fibonacci-Hessenberg matrices and their generalizations satisfy this property. Simple systems of linear equations are given whose solutions are Fibonacci fractions.

The Fibonacci dimension fdim(G) of a graph G was introduced in [1] as the smallest integer d such that G admits an isometric embedding into Γd, the d-dimensional Fibonacci cube. The Fibonacci dimension of the resonance graphs of catacondensed benzenoid systems is studied. This study is inspired by the fact, that the Fibonacci cubes are precisely the resonance graphs of a subclass of the catacon...

Here, we investigate the Fibonacci numbers whose sum of aliquot divisors is also a Fibonacci number (the prime Fibonacci numbers have this property).

Among numerical sequences, the Fibonacci numbers Fn have achieved a kind of celebrity status. Indeed, Koshy gushingly refers to them as one of the “two shining stars in the vast array of integer sequences” [16, p. xi]. The second of Koshy’s “shining stars” is the Lucas numbers, a close relative of the Fibonacci numbers, about which we will say more below. The Fibonacci numbers are famous for po...

1. Robert P. Backstrom. "On the Determination of the Zeros of the Fibonacci Sequence." The Fibonacci Quarterly 4, No. 4 (1966):313-322. 2. R. D. Carmichael. "On the Numerical Factors of the Arithmetic Forms a + 3." Annals of Mathematics9 2nd Ser. 15 (1913):30-70. 3. John H. Halton. "On the Divisibility Properties of Fibonacci Numbers." The Fibonacci Quarterly 4, No. 3 (1966):217-240. 4. D.H. Le...

For integers s,k with s 0 and k 0 , we define a class of lower triangular Toeplitz matrices U (s,k) n of type (s,k) , whose non-zero entries are the classical Horadam numbers U (a,b) i . In this paper, we derive a convolution formula containing the Horadam numbers. Using this formula, we obtain several combinatorial identities involving the Horadam numbers and the generalized Fibonacci numbers....

An !-sequence is defined by an = !an−1 − an−2, with initial conditions a0 = 0, a1 = 1. These !-sequences play a remarkable role in partition theory, allowing !generalizations of the Lecture Hall Theorem and Euler’s Partition Theorem. These special properties are not shared with other sequences, such as the Fibonacci sequence, defined by second-order linear recurrences. The !-sequence gives rise...

In this paper, we consider the usual and generalized order-k Fibonacci and Pell recurrences, thenwe define a new recurrence, which we call generalized order-k F–P sequence. Also we present a systematic investigation of the generalized order-k F–P sequence. We give the generalized Binet formula, some identities and an explicit formula for sums of the generalized order-k F–P sequence by matrix me...

In the language of mathematical chemistry, Fibonacci cubes can be defined as the resonance graphs of fibonacenes. Lucas cubes form a symmetrization of Fibonacci cubes and appear as resonance graphs of cyclic polyphenantrenes. In this paper it is proved that the Wiener index of Fibonacci cubes can be written as the sum of products of four Fibonacci numbers which in turn yields a closed formula f...

The Fibonacci dimension fdim(G) of a graph G was introduced in [7] as the smallest integer d such that G admits an isometric embedding into Qd, the d-dimensional Fibonacci cube. A somewhat new combinatorial characterization of the Fibonacci dimension is given, which enables more comfortable proofs of some previously known results. In the second part of the paper the Fibonacci dimension of the r...