The recent introduction of generalized Appell sequences in the framework of Clifford Analysis solved an open question about a suitable construction of power-like monogenic polynomials as generalizations of the integer powers of a complex variable. The deep connection between Appell sequences and Pascal’s triangle called also attention to other number triangles and, at the same time, to the cons...

so that there are infinitely many numbers occurring at least 4 times in the Pascal Triangle. Stripped of these trivialities, the more interesting problem becomes to determine the natural numbers that occur at least twice as binomial coefficients of the shape ( n k) with 2 k 1 2n, and this is yet unsolved in its full generality. The only nontrivial solutions known at this time are the following:...

The summation formula within Pascal triangle resulting in the Fi-bonacci sequence is extended to the q-binomial coefficients q-Gaussian triangles [1, 2]. 1 Pisa historical remark The Fibonacci sequence origin is attributed and referred to the first edition (lost) of " Fiber abaci " (1202) by Leonardo Fibonacci [Pisano] (see second edition from 1228 reproduced as Il Liber Abaci di Leonardo Pisan...

An interval vector is a (0, 1)-vector in Rn for which all the 1’s appear consecutively, and an interval-vector polytope is the convex hull of a set of interval vectors in Rn. We study three particular classes of interval vector polytopes which exhibit interesting geometric-combinatorial structures; e.g., one class has volumes equal to the Catalan numbers, whereas another class has face numbers ...

In this paper, we partially solve an open problem, due to J. C. Molluzzo in 1976, on the existence of balanced Steinhaus triangles modulo a positive integer n, that are Steinhaus triangles containing all the elements of Z/nZ with the same multiplicity. For every odd number n, we build an orbit in Z/nZ, by the linear cellular automaton generating the Pascal triangle modulo n, which contains infi...

BACKGROUND The aim of this study was to analyze the pattern and types of sensory nerve endings in ankle ligaments using immunohistochemical techniques, in order to gain more insight into functional ankle stability. METHODS One hundred forty ligaments from 10 cadaver feet were included: the calcaneofibular and anterior/posterior talofibular ligaments from the lateral complex; inferior extensor...

The relation between the technique of conformal flat and Damour-Ruffini-Zhao's method is investigated in this paper. It is pointed out that the two methods give the same results when the metric has the form g αβ=0 , with α = 0, 1 and β = 2, 3. It is indicated that the two methods are not equivalent for general case.

A binary Steinhaus triangle is a of zeroes and ones that points down with the same local rule as Pascal modulo 2. said to be rotationally symmetric, horizontally symmetric or dihedrally if it invariant under 120 degrees rotation, horizontal reflection both, respectively. The first part this paper devoted study linear subspaces triangles. We obtain simple explicit bases for each them by using el...