A tile composed of two pieces, which we refer to as a fence tile, is introduced to give the Tribonacci numbers a new combinatorial interpretation. The interpretation is used to construct a Pascal-like triangle and various identities concerning the triangle are proven. An intuitive proof of a general identity for the Tribonacci numbers in terms of sums of products of the binomial coefficients is...

In this paper we generalize Pascal’s Triangle and examine the connections between the generalized triangles and powering integers and polynomials respectively. The interesting and really romantic Pascal’s Triangle is a favourite research field of mathematicians for a very long time. The table of binomial coefficients has been named after Blaise Pascal, a French scientist, but was known already ...

It is well known that striking patterns of triangles can be produced from Pascal's triangle by replacing each binomial coefficient by its residue with respect to a certain modulus. The arrays thus produced were considered by various authors; see, for instance, Gould [5], Gardner [1], Long [10], or Sved [17]. For example, Pascal's triangle mod 2 (Fig. 1) is the array of zeros and ones obtained b...

where e, h are both nonnegative and dn,k = 0 unless n ≥ k ≥ 0. We call D(e, h) the Delannoy-like triangle. The entries dn,k count lattice paths from (0, 0) to (n − k, k) using the steps (0, 1), (1, 0) and (1, 1) with assigned weights 1, e, and h. Some wellknown combinatorial triangles are such matrices, including the Pascal triangle D(1, 0), the Fibonacci triangle D(0, 1), and the Delannoy tria...

Two matrix equations, Pascal matrix equation and inverse Pascal matrix, are derived and demonstrated in this work. These matrix equations are used for making a conversion and inversion between the coefficients of an analog low pass filter transfer function H(s) and a digital transfer function H(z). The involving of the Pascal’s triangle in both matrix equations is helpful, easier to transform b...

A new set of fractal multiband antennas called mod Sierpinski gaskets is presented.Mod Sierpinski fractal antennas derive from the Pascal triangle and present a log-periodic behavior, which is a consequence of their self-similarity properties.Mod Sierpinski fractal antennas constitute a generalization of the classical Sierpinski antenna.

A lattice path enumeration approach is exploited in order to derive a binomial summation formula for the number of paths in the lattice (0, 1, . . . , d) with start 0 and n steps. This approach reveals multisection style binomial summation identitities and, particularly, a novel relationship between Fibonacci numbers and the Pascal triangle. Mathematics Subject Classification: 05A10, 11B39

The periodontal ligament receives a rich sensory nerve supply and contains many nociceptors and mechanoreceptors. Although its various kinds of mechanoreceptors have been reported in the past, only recently have studies revealed that the Ruffini endings--categorized as low-threshold, slowly adapting, type II mechanoreceptors--are the primary mechanoreceptors in the periodontal ligament. The per...

The binomial interpolated transform of a sequence is a generalization of the wellknown binomial transform. We examine a Pascal-like triangle, on which a binomial interpolated transform works between the left and right diagonals, focusing on binary recurrences. We give the sums of the elements in rows and in rising diagonals, and we define two special classes of these arithmetical triangles.