Fibonacenes (zig-zag unbranched catacondensed benzenoid hydrocarbons) are a class of polycyclic conjugated systems whose molecular graphs possess remarkable properties, often related with the Fibonacci numbers. This article is a review of the chemical graph theory of fibonacenes, with emphasis on their Kekulé–structure–related and Clar–structure–related properties. ————————————————— ∗Supported ...

In [3], H. Belbachir and F. Bencherif generalize to bivariate polynomials of Fibonacci and Lucas, properties obtained for Chebyshev polynomials. They prove that the coordinates of the bivariate polynomials over appropriate basis are families of integers satisfying remarkable recurrence relations. [7], Mario Catalani define generalized bivariate polynomials, from which specifying initial conditi...

for all nonnegative integers i, j such that j ≤ i, as illustrated in Figure 1.1. The points in R2 associated with ( i j ) , ( i+1 j ) , and ( i+1 j+1 ) form a unit equilateral triangle. This arrayal is called the natural arrayal of Pascal’s triangle in R2. For all t ∈ R : −√3 < t < √3 and nonnegative integers k, define k(t) to be the sum of all binomial coefficients associated with points in R2...

The extended Fibonacci sequence of numbers and polynomials is introduced and studied. The generating function, recurrence relations, an expansion in terms of multinomial coefficients, and several properties of the extended Fibonacci numbers and polynomials are obtained. Interesting relations between them and probability problems which take into account lengths of success and failure runs are al...

Considering the five periods and six qi's theory in TCM almost shares a common basis of stem-branch system with the five elements of containing notes, studying the principle or mathematical structure behind the five elements of containing notes can surely bring a novel view for the five periods and six qi's researches. By analyzing typical mathematical rules included in He tu, Luo shu, and stem...

In a sequence of generalized Fibonacci trees, the kth tree has the (k c(i))th tree as its ith subtree for a nondecreasing sequence of positive integers c(i), i = 1; :::; r. For particular initializations, each tree in the generalized Fibonacci sequence solves a minimax coding problem related to Varn coding. Speci cally, each symbol from a uniformly distributed source is to be encoded by a strin...

Generalizations of Binet’s theorem are used to produce generalized Pell sequences from two families of silver means. These Pell sequences are also generated from the family of Fibonacci polynomials. A family of Pell-Lucas sequences are also generated from the family of Lucas polynomials and from another generalization of Binet’s formula. A periodic set of cyclic constants are generated from the...

Let a be a nonnegative real number and define a quasi-Fibonacci polynomial sequence by F a 1 (x) = −a, F a 2 (x) = x − a, and F a n (x) = F a n−1(x) + xF a n−2(x) for n ≥ 2. Let ra n denote the maximum real root of F a n . We prove for certain values of a that the sequence {ra 2n} converges monotonically to βa = a 2 + a from above and the sequence {ra 2n+1} converges monotonically to βa from be...

To parallelize applications that require the use of random numbers, an efficient and good quality parallel random number generator is required. In this paper, we study the parallelization of lagged Fibonacci generators for distributed memory parallel computers. Two popular ways of generating a random sequence in parallel are studied: the contiguous subsequence technique and the leapfrog techniq...