TY - GEN AU - Liptai, Kálmán AU - Németh, L AU - Szakács, Tamás AU - Szalay, László TI - On certain Fibonacci representations PY - 2024 UR - https://m2.mtmt.hu/api/publication/34762407 ID - 34762407 LA - English DB - MTMT ER - TY - JOUR AU - Győry, Kálmán AU - Pethő, Attila AU - Szalay, László TI - Decomposable Forms Generated by Linear Recurrences JF - JOURNAL OF INTEGER SEQUENCES J2 - J INTEGER SEQ VL - 27 PY - 2024 IS - 3 PG - 19 SN - 1530-7638 UR - https://m2.mtmt.hu/api/publication/34742592 ID - 34742592 AB - Consider k ≥ 2 distinct, linearly independent, homogeneous linear recurrences of order k satisfying the same recurrence relation. We prove that the recurrences are related to a decomposable form of degree k, and there is a general identity with a suitable exponential expression depending on the recurrences. This identity is a common and very broad generalization of several known identities. Further, if the recurrences are integer sequences, then the diophantine equation associated with the decomposable form and the exponential term has infinitely many integer solutions generated by the terms of the recurrences. We describe a method for the complete factorization of the decomposable form. Both the form and its decomposition are explicitly given if k = 2, and we present a typical example for k = 3. The basic tool we use is the matrix method. © 2024, University of Waterloo. All rights reserved. LA - English DB - MTMT ER - TY - JOUR AU - Liptai, Kálmán AU - Szalay, László TI - Distribution Generated by a Random Inhomogenous Fibonacci Sequence JF - MEDITERRANEAN JOURNAL OF MATHEMATICS J2 - MEDITERR J MATH VL - 21 PY - 2024 IS - 1 PG - 17 SN - 1660-5446 DO - 10.1007/s00009-023-02563-3 UR - https://m2.mtmt.hu/api/publication/34538292 ID - 34538292 N1 - Funding Agency and Grant Number: Hungarian Scientific Research Fund [128088, 130909]; Hungarian National Foundation for Scientific Research Grant [VEGA 1/0776/21]; Slovak Scientific Grant Agency Funding text: For L. Szalay, this research was supported by the Hungarian National Foundation for Scientific Research Grant No. 128088, and No. 130909, and the Slovak Scientific Grant Agency VEGA 1/0776/21. AB - Let G_0=0 G 0 = 0 and G_1=1 G 1 = 1 . The present study deals with the inhomogeneous version \begin{aligned} G_n=G_{n-1}+G_{n-2}+w_{n-2} \end{aligned} G n = G n - 1 + G n - 2 + w n - 2 of the Fibonacci sequence, where w_{n-2} w n - 2 takes value a with probability p , and does value b with 1-p 1 - p . We describe the probability distribution of the values of G_n G n with fixed n , and examine the properties like expected value and variance. The most challenging feature is the fractal-like structure of the distribution. LA - English DB - MTMT ER - TY - JOUR AU - Major, Laszlo AU - Németh, László AU - Pahikkala, Anna AU - Szalay, László TI - Self-avoiding walks of specified lengths on rectangular grid graphs JF - AEQUATIONES MATHEMATICAE J2 - AEQUATIONES MATH VL - 98 PY - 2024 SP - 215 EP - 239 PG - 25 SN - 0001-9054 DO - 10.1007/s00010-023-00977-8 UR - https://m2.mtmt.hu/api/publication/34114194 ID - 34114194 AB - The investigation of self-avoiding walks on graphs has an extensive literature. We study the notion of wrong steps of self-avoiding walks on rectangular shape n x m grids of square cells (Manhattan graphs) and examine some general and special cases. We determine the number of self-avoiding walks with one and with two wrong steps in general. We also establish some properties, like unimodality and sum of the rows of the Pascal-like triangles corresponding to the walks. We also present particular recurrence relations on the number of self-avoiding walks on the n x 2 grids with any specified number of wrong steps. LA - English DB - MTMT ER - TY - JOUR AU - Mufid, M.S. AU - Szalay, László TI - Interpolation polynomials associated to linear recurrences JF - TURKISH JOURNAL OF MATHEMATICS J2 - TURK J MATH VL - 47 PY - 2023 IS - 7 SP - 1932 EP - 1943 PG - 12 SN - 1300-0098 DO - 10.55730/1300-0098.3473 UR - https://m2.mtmt.hu/api/publication/34410236 ID - 34410236 AB - Assume that (Gn)n∈Z is an arbitrary real linear recurrence of order k. In this paper, we examine the classical question of polynomial interpolation, where the basic points are given by (t, Gt) (n0 ≤ t ≤ n1 ). The main result is an explicit formula depends on the explicit formula of Gn and on the finite difference sequence of a specific sequence. It makes it possible to study the interpolation polynomials essentially by the zeros of the characteristic polynomial of (Gn). During the investigations, we developed certain formulae related to the finite differences. © TÜBİTAK LA - English DB - MTMT ER - TY - CHAP AU - Szalay, László ED - Gharib, Gharib ED - Merker, Jochen ED - Qazza, Ahmad ED - Burqan, Aliaa ED - Cortés, Juan C. ED - Zeidan, Dia TI - Explicit Formulae of Linear Recurrences T2 - Mathematics and Computation PB - Springer Nature Singapore CY - Singapore SN - 9789819904471 T3 - Springer Proceedings in Mathematics & Statistics, ISSN 2194-1009 ; 418. PY - 2023 SP - 277 EP - 284 PG - 8 DO - 10.1007/978-981-99-0447-1_23 UR - https://m2.mtmt.hu/api/publication/34183792 ID - 34183792 AB - One important and widely studied problem in the theory of linear recurrences is to find explicit formulae for the general term of the sequences. Having an explicit formula facilitates the research of the properties of the sequence we investigate. The main tool is to apply the fundamental theorem of homogeneous linear recurrences, but other approaches may work as well. In the present paper, we concentrate on a specific case when the characteristic polynomial of the sequence has a double zero, and on a general formula. LA - English DB - MTMT ER - TY - CHAP AU - Németh, László AU - Szalay, László ED - Gharib, Gharib ED - Merker, Jochen ED - Qazza, Ahmad ED - Burqan, Aliaa ED - Cortés, Juan C. ED - Zeidan, Dia TI - Generalizations of the Fibonacci Sequence with Zig-Zag Walks T2 - Mathematics and Computation PB - Springer Nature Singapore CY - Singapore SN - 9789819904471 T3 - Springer Proceedings in Mathematics & Statistics, ISSN 2194-1009 ; 418. PY - 2023 SP - 303 EP - 310 PG - 8 DO - 10.1007/978-981-99-0447-1_26 UR - https://m2.mtmt.hu/api/publication/34183753 ID - 34183753 AB - The examination of the recurrence sequences associated with combinatorial constructions has been very extensive in the last decades. One of the most famous recurrence sequences is the Fibonacci sequence. We give two digraph constructions defined on the hyperbolic and on the Euclidean square mosaics, respectively, and we introduce two zig-zag type walks associating to the Fibonacci and its generalized sequences. Then we determine the recurrence relations and we give some examples. LA - English DB - MTMT ER - TY - JOUR AU - Liptai, Kálmán AU - Szalay, László TI - Random inhomogeneous binary recurrences JF - ANNALES UNIVERSITATIS SCIENTIARUM BUDAPESTINENSIS DE ROLANDO EOTVOS NOMINATAE SECTIO COMPUTATORICA J2 - ANN UNIV SCI BP R EÖTVÖS NOM SECT COMPUT VL - 54 PY - 2023 SP - 253 EP - 263 PG - 11 SN - 0138-9491 UR - https://m2.mtmt.hu/api/publication/34183473 ID - 34183473 LA - English DB - MTMT ER - TY - JOUR AU - Belbachir, Hacène AU - Rami, Fella AU - Németh, László AU - Szalay, László TI - A generalization of the hyperbolic Pascal pyramid JF - INDIAN JOURNAL OF PURE & APPLIED MATHEMATICS J2 - INDIAN J PURE AP MAT PY - 2023 PG - 11 SN - 0019-5588 DO - 10.1007/s13226-023-00481-4 UR - https://m2.mtmt.hu/api/publication/34177993 ID - 34177993 LA - English DB - MTMT ER - TY - JOUR AU - Filipin, A. AU - Szalay, László TI - TRIANGULAR DIOPHANTINE TUPLES FROM {1, 2} JF - RAD HRVATSKE AKADEMIJE ZNANOSTI I UMJETNOSTI-MATEMATICKE ZNANOSTI J2 - RAD HRVAT AKAD ZNAN UMJET MAT ZNAN VL - 27 PY - 2023 IS - 555 SP - 55 EP - 70 PG - 16 SN - 1845-4100 DO - 10.21857/ygjwrcp48y UR - https://m2.mtmt.hu/api/publication/34131814 ID - 34131814 AB - In this paper, we prove that there does not exist a set of four positive integers {1, 2, c, d} such that a product of any two of them increased by 1 is a triangular number. © 2023, Croatian Academy of Sciences and Arts. All rights reserved. LA - English DB - MTMT ER -