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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.

Several articles deal with tilings with squares and dominoes on 2-dimensional boards, but only a few on boards in 3-dimensional space. We examine a tiling problem with coloured cubes and bricks on ...

The Exeter point of a given triangle ABC is the center of perspective of the tangential triangle and the circummedial triangle of the given triangle. The process of the Exeter point from the centroid serves as a base for defining the Exeter transformation with respect to the triangle ABC, which maps all points of the plane. We show that a point, its image, the symmedian, and three exsymmedian points of the triangle are on the same conic. The Exeter transformation of a general line is a fourth-order curve passing through the exsymmedian points. We show that each image point can be the Exeter transformation of four different points. We aim to determine the invariant lines and points and some other properties of the transformation.

Consider the diophantine equation $(3^{x_1}-1)(3^{x_2}-1)=(5^{y_1}-1)(5^{y_2}-1)$ in positive integers $x_1\le x_2$, and $y_1\le y_2$. Each side of the equation is a product of two terms of a given binary recurrence, respectively. In this paper, we prove that the only solution to the title equation is $(x_1,x_2,y_1,y_2)=(1,2,1,1)$. The main novelty of our result is that we allow products of two terms on both sides., Comment: 10 pages, accepted for publication

We propose an extension to the work of Lucca [Giovanni Lucca, Integer sequences and circle chains inside a hyperbola, Forum Geometricorum, Volume 19. 2019, 11–16]. Our goal is to examine chains of ellipses inside a hyperbola, and we derive recurrence relations of centers and minor (major) axes of the ellipse chains. We also determine conditions for these recurrence sequences to consist of integer numbers.

Several articles deal with tilings with squares and dominoes of the well-known regular square mosaic in Euclidean plane, but not any with the hyperbolic regular square mosaics. In this article, we examine the tiling problem with colored squares and dominoes of one type of the possible hyperbolic generalization of $(2\times n)$-board., Comment: 10 pages, 8 figures

Let $F_n$ denote the $n^{th}$ term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation $F_1^p+2F_2^p+\cdots+kF_{k}^p=F_{n}^q$ in the positive integers $k$ and $n$, where $p$ and $q$ are given positive integers. A complete solution is given if the exponents are included in the set $\{1,2\}$. Based on the specific cases we could solve, and a computer search with $p,q,k\le100$ we conjecture that beside the trivial solutions only $F_8=F_1+2F_2+3F_3+4F_4$, $F_4^2=F_1+2F_2+3F_3$, and $F_4^3=F_1^3+2F_2^3+3F_3^3$ satisfy the title equation., Comment: 12 pages

The hyperbolic Pascal triangle ${\cal HPT}_{4,q}$ $(q\ge5)$ is a new mathematical construction, which is a geometrical generalization of Pascal's arithmetical triangle. In the present study we show that a natural pattern of rows of ${\cal HPT}_{4,5}$ is almost the same as the sequence consisting of every second term of the well-known Fibonacci words. Further, we give a generalization of the Fibonacci words using the hyperbolic Pascal triangles. The geometrical properties of a ${\cal HPT}_{4,q}$ imply a graph structure between the finite Fibonacci words., Comment: 10 pages, 4 figures, Acta Univ. Sapientiae, Mathematica, 2017

In this article, we study the structure of the graph implied by a given map on the set $S_p=\{1,2,\dots,p-1\}$, where $p$ is an odd prime. The consecutive applications of the map generate an integer sequence, or in graph theoretical context a walk, that is linked to the discrete logarithm problem., Comment: 6 pages, 2 figures

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 quantitative properties such as the number, the sum, and the alternating sum of the elements of a row. Moreover, the pattern of the rows, and the appearence of some binary recurrences in a fixed hyperbolic triangle are investigated., Comment: 18 pages, 15 figures

In the 3-dimensional hyperbolic spaceH3there are four regular mosaics with asymptotic regular pyramids and each is related to a reciprocal regular mosaic of ideal vertices. This paper deals with the connections of the numbers of the elements of the belts surrounding a base point and the limits of their ratios.