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In this paper, we introduce a new generalization of Pascal's triangle. The
new object is called the hyperbolic Pascal triangle since the mathematical
background goes back to regular mosaics on the hyperbolic plane. We describe
precisely the procedure of how to obtain a given type of hyperbolic Pascal
triangle from a mosaic. Then we study certain qu...
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Context 1
... q} = {4, 4}). Then the subgraph G P with its labelling returns with Pascal's original triangle (see Figure 2). The other two Euclidean regular mosaics have no great interest since they are also as- sociated with Pascal's triangle (see Figure 3). ...
Similar publications
Recently, a new generalization of Pascal's triangle, the so-called hyperbolic Pascal triangles were introduced. The mathematical background goes back to the regular mosaics in the hyperbolic plane. In this article, we investigate the paths in the hyper-bolic Pascal triangle corresponding to the regular mosaic {4, 5}, in which the binary recursive s...
A new generalization of Pascal's triangle, the so-called hyperbolic Pascal triangles
were introduced in [1]. The mathematical background goes back to the regular mosaics
in the hyperbolic plane. The alternating sum of elements in the rows was given in the
special case {4,5} of the hyperbolic Pascal triangles. In this article, we determine the
alter...
In this paper we introduce a new type of Pascal's pyramids. The new object is
called hyperbolic Pascal pyramid since the mathematical background goes back to
the regular cube mosaic (cubic honeycomb) in the hyperbolic space. The
definition of the hyperbolic Pascal pyramid is a natural generalization of the
definition of hyperbolic Pascal triangle a...
Citations
... The difficulty that may arise in practice is finding the zeros precisely. For the non-homogeneous version of (2), see [2] (with two sequences) and [5] (generally). The crucial point of our approach is to transform (1) in order to obtain a corresponding vector recurrence in the form (2) (which will be given later in (6)). ...
... The notion of the hyperbolic Pascal triangle (see [2]) was introduced a few years ago as a variation of Pascal's arithmetic triangle. The novelty was to extend the link between the infinite graph corresponding to the regular square Euclidean mosaic and classical Pascal's triangle to the hyperbolic plane, which contains infinitely many types of regular mosaics. ...
... Belbachir et al. [2] discovered several properties of HPT's, we recall here only one result. The sum of the elements in row n gives the number of shortest paths ν n from the base vertex to row n. ...
Solution of various combinatorial problems often requires vector recurrences of higher order (i.e., the order is larger than 1). Assume that there are given matrices A1, A2,..., As, all from C k×k. These matrices allow us to define the vector recurrence v(n) = A1v(n−1)+ A2v(n−2) + · · · + Asv(n−s) for the vectorsvn ∈ C k , n ≥ s. The paramount result of this paper is that we could separate the component sequences of the vectors and find a common linear recurrence relation to describe them. The principal advantage of our approach is a uniform treatment and the possibility of automatism. We could apply the main result to answer a problem that arose concerning the rows of the modified hyperbolic Pascal triangle with parameters {4, 5}. We also verified two other statements from the literature in order to illustrate the power of the method.
... Figure 2 shows the graph of the first four layers of GHPP when {α n } = {β n } = {γ n } = {F n }, the Fibonacci sequence. And Figures 3,4, and 5 illustrate the connection between levels two, three, four and five in it. ...
In the present paper, we consider a variation of the hyperbolic Pascal pyramid where the three leg-sequences (the constant 1 sequence) are replaced by the sequences \(\lbrace \alpha _n\rbrace _{n\ge 0}\), \(\lbrace \beta _n\rbrace _{n\ge 0}\) and \(\lbrace \gamma _n\rbrace _{n\ge 0}\) with \(\alpha _0=\beta _0=\gamma _0=\Omega \), and we describe the values of elements. Then we give the recurrence relations associated to the sums of the values on levels in the generalized hyperbolic Pascal’s pyramids. The order of these recurrences is six.
... Each regular mosaic induces a so-called hyperbolic Pascal triangle, following and generalizing the connection between the classical Pascal's triangle and the Euclidean regular square mosaic {4, 4}. For more details see [2], but here we also collect some necessary information. ...
... Let s n denotes the number of the vertices in row n. Note that the sequence {s n } is described in [2]. ...
... with the initial values a 1 = 0, a 2 = 1, a 3 = 2; b 1 = 0, b 2 = 0, b 3 = q − 4; s 1 = 2, s 2 = 3, s 3 = q (see [2]). ...
Recently a new generalization of Pascal's triangle, the family of so-called hyperbolic Pascal triangles (in short HPT) was introduced. In this paper, we consider a variation of HPT by replacing the two leg-sequences of the triangle by arbitrary sequences {αn}n≥0 and {βn}n≥0. Originally the legs were the constant 1 sequence. We describe some quantitative properties via recurrences relations, explicit formulae and generating functions. In addition, we present a useful general result on linear recurrences sequences diverted by an arbitrary sequence.
... Németh és Szalay [19]-ben összefoglaló jelleggel egy újabb típusú általánosítást, az un. hiperbolikus Pascal háromszögeket [2] mutatta be, amelyről egy rövid angol nyelvű áttekintés található a [3] összefoglalóban. További tulajdonságok a [2,12,[16][17][18] tudományos munkákban részletesen megtalálhatók. ...
... hiperbolikus Pascal háromszögeket [2] mutatta be, amelyről egy rövid angol nyelvű áttekintés található a [3] összefoglalóban. További tulajdonságok a [2,12,[16][17][18] tudományos munkákban részletesen megtalálhatók. ...
Pascal's triangle is a triangular arrangement of the binomial coefficients , which is well-known and used in several fields of mathematics. In our review we introduce its special higher dimensional generalizations, the so-called Pascal tetrahedron, the Pascal simplex, and an interesting version of the tetrahedron in the space H 2 ×R. We give the growings of layers by layers with recurrence relations.
... In number theory the study of recursive sequences and related identities by binomial coefficients has a certain relevance (see [1,2,7,17,19,[21][22][23]30,32,33,37], and references therein). In particular, many authors have studied the connections between special families of recursive sequences and generalized Pascal's triangles (for a rich compendium see [8][9][10][11][12]25,26] and [36]). ...
... Relation (9) and the identities (10), (12), (13) and (14) show the relation (7) of the statement. □ ...
... Our aim is to show that for every k > 0 the sequence S k,m (see (9) and (7)) referred to generalized Pascal's triangle is a k-Padovan-like sequence. ...
One of the most interesting properties of Pascal’s triangle is that the sequence of the sums of the elements on its diagonals is the best known recurrence sequence, the Fibonacci sequence. It is also known that other diagonals can be associated with other relevant recurrence sequences, such as the Padovan and k-Padovan sequences. In this paper, we see that similar properties also hold for diagonals of generalized Pascal’s triangles. We show that the diagonal sums in generalized Pascal’s triangles belong to the family of the so-called ‘k-Padovan-like sequences’ which are linear recurrences of order k with constant coefficients. A recurrence connection between the k-Padovan and k-Padovan-like sequences is derived.
... This article proceeds our work on the variants of Pascal's arithmetic triangle for which there are several approaches to generalize or extend (see, for instance, [4]). One recent variation is called hyperbolic Pascal triangles, see [3], where we described how to construct them as we follow and generalize the connection between the classical Pascal triangle and the Euclidean regular square mosaic. The principal idea behind is to leave the Euclidean plane because in the hyperbolic plane there exist infinitely many regular mosaics. ...
... Returning to the hyperbolic Pascal triangle linked to mosaic {4, 5} one finds the Fibonacci sequence (A000045 in the OEIS [9]) by going along a specific zig-zag path. The authors proved in [3,7] that all the integer linear homogeneous recurrence sequences ...
We define a so-called square $k$-zig-zag shape as a part of the regular square grid. Considering the shape as a $k$-zig-zag digraph, we give values of its vertices according to the number of the shortest paths from a base vertex. It provides several integer sequences, whose higher-order homogeneous recurrences are determined by the help of a special matrix recurrence.
... Most a Pascal háromszög egy újabb, a közelmúltban felfedezett általánosítását mutatjuk be összefoglaló jelleggel, amelyről egy rövid angol nyelvű áttekintés található a [6] publikációban. Mivel a klasszikus Pascal háromszög elemei egy négyzetrács csúcspontjaira is írhatók és az értékek egy kitüntetett csúcsponttól való legrövidebb rácsútvonalak számát adják, ezért ha az euklideszi négyzetrácsot kicseréljük hiperbolikus szabályos rácsra, vagy mozaikra, akkor a Pascal háromszög egy újabb változatát, a hiperbolikus Pascal háromszögeket kapjuk [5]. ...
... Kapcsolatot keresünk a hiperbolikus Pascal háromszög sorainak egyfajta mintázata és a jól ismert bináris rekurzív sorozat, a Fibonacci szavak sorozata között. Továbbá megmutatjuk, hogy ha egy bizonyos hiperbolikus Pascal háromszöget irányított gráf formában adjuk meg, akkor a másodrendű rekurzív sorozatok egy jól definiálható osztályának elemei a gráfban egy-egy útvonal mentén megtalálhatók [5,27,[34][35][36]. ...
... A HPT 4,q növekedési algoritmusa 1. Tétel. [5]. Az {a n }, {b n } és {s n } sorozatok az ...
Pascal’s triangle is a triangular arrangement of binomial coefficient, which is well known and used in several fields of mathematics. The family of hyperbolic Pascal triangles is a recently introduced variation among different kinds of generalizations and extentions. In this paper, we give a survey on hyperbolic Pascal triangles. Some properties have analogue in Pascal’s classical triangle, but a few new phenomena have also been appeared.
... Pascal triangle (known as the Yang Hui Triangle in China) is a triangular arrangement of the binomial coefficients and that it contains many remarkable numerical relations [19,64]. Pascal-Triangle has also become a suitable model system for the exploration of statistical mechanical structures [53,64]. ...
PPascal triangle (known as Yang Hui Triangle in Chinese) is an important model in mathematics while the entropy has been heavily studied in physics or as uncertainty measure in information science. How to construct the the connection between Pascal triangle and uncertainty measure is an interesting topic. One of the most used entropy, Tasllis entropy, has been modelled with Pascal triangle. But the relationship of the other entropy functions with Pascal triangle is still an open issue. Dempster-Shafer evidence theory takes the advantage to deal with uncertainty than probability theory since the probability distribution is generalized as basic probability assignment, which is more efficient to model and handle uncertain information. Given a basic probability assignment, its corresponding uncertainty measure can be determined by Deng entropy, which is the generalization of Shannon entropy. In this paper, a Pseudo-Pascal triangle based the maximum Deng entropy is constructed. Similar to the Pascal triangle modelling of Tasllis entropy, this work provides the a possible way of Deng entropy in physics and information theory.
... In this paper, we pose, and investigate the analogous question in Pascal pyramid, in one extension of Pascal's triangle. There exist other generalizations and extensions of Pascal's arithmetical triangle (see, for instance [5,4,10,11]), and if one defines diagonals, in other words rays, generally the question of unimodality, the question of the sum of the elements included arise. Let us quote, in case of unimodality the article of Mező [12] (Stirling triangle), Corcino et al [8] (Whitney triangle), Tebtoub [15] (regular Lah triangle), Belbachir, Tebtoub [6] (2-successive associated Stirling numbers). ...
... Let us quote, in case of unimodality the article of Mező [12] (Stirling triangle), Corcino et al [8] (Whitney triangle), Tebtoub [15] (regular Lah triangle), Belbachir, Tebtoub [6] (2-successive associated Stirling numbers). Concerning the problem of sums we cite the work of Belbachir, Komatsu and Szalay [3] (generalized Pascal triangle), Ahmia and Szalay [1] (weighted sums), Belbachir, Németh and Szalay [4] (hyperbolic Pascal triangles). ...
In this paper, we describe the recurrence relation associated to the sum of diagonal elements lying along a finite ray of certain type crossing the three-dimensional Pascal pyramid. Then we determine the corresponding generating function. Finally, we deal with the so-called Morgan–Voyce phenomenon, and applying the new arguments we prove an identity, which makes it possible to prove several recurrence relations conjectured in Sloane's On-Line Encyclopedia of Integer Sequences.
... The fundamental work of Poincare [28] has given base for the research of Fuchsian groups, that is, discrete transformations groups of the Lobachevskii plane 2 , and has generated thereby a particular interest to the "hyperbolic mosaics". This subject still attracts a lot of attention (see, for instance, [3,4,16,26,27]). There are also other geometrical objects closely connected with Fuchsian groups, in particular, Coxeter groups of the plane 2 (see, for instance, [8,18,20]). ...
A hyperbolic plane O H of positive curvature can be realized on the exterior domain with respect to an oval curve in a projective plane. The plane O H is the Cayley-Klein model of a two-dimensional de Sitter space. It is proved that there is no regular mosaic on the plane O H .
