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Öğe Balancing Diophantine triples with distance 1(SPRINGER, 2015) Alp, Murat; Irmak, Nurettin; Szalay, LaszloFor a positive real number let the Balancing distance be the distance from to the closest Balancing number. The Balancing sequence is defined by the initial values , and by the binary recurrence relation , . In this paper, we show that there exist only one positive integer triple such that the Balancing distances , and all are exactly 1.Öğe BALANCING WITH POWERS OF THE LUCAS SEQUENCE OF RECURRENCE un = Aun-1 - un-2(Univ Miskolc Inst Math, 2017) Irmak, Nurettin; Alp, MuratIn this paper, we show that there is no solution of the diophantine equation u(1)(k) + u(2)(k) + . . . + u(n+1)(l) + u(n+2)(l) + . . . + u(n+r)(l) for special cases of k and l where the elements of sequence {u(n)} satisfy the relation u(n) = Aun-1 - u(n-2) with u(0) = 0, u(1) = 1 and A >= 3 is a positive integer.Öğe Cat1-polygroups and pullback cat1-polygroups(Iranian Mathematical Society, 2014) Davvaz, Bijan; Alp, MuratIn this paper, we give the notions of crossed polymodule and cat1-polygroup as a generalization of Loday's definition. Then, we define the pullback cat1-polygroup and we obtain some results in this respect. Specially, we prove that by a pullback cat1- polygroup we can obtain a cat1-group. © 2014 Iranian Mathematical Society.Öğe Crossed modules of hypergroups associated with generalized actions(HACETTEPE UNIV, FAC SCI, 2016) Alp, Murat; Davvaz, BijanIn this article, by using the notion of generalized action, we introduce the concept of crossed module of hypergroups, in the sense of Marty, and its related structures from the light of crossed polymodules. Hyper-groups in the sense of Marty are more different than polygroups since they have not identity element or inverse element in general. Examples of crossed modules of hypergroups are originally presented. These examples illustrate the structure and behavior of crossed modules of hypergroups. Moreover, we obtain a crossed module in the sense of Whitehead from a crossed module of hypergroups by applying the notion of fundamental relation.Öğe CROSSED POLYMODULES AND FUNDAMENTAL RELATIONS(UNIV POLITEHNICA BUCHAREST, SCI BULL, 2015) Alp, Murat; Davvaz, BijanIn this paper, we introduce the notion of crossed polymodule of polygroups and we give some of its properties. Our results extend the classical results of crossed modules to crossed polymodules. One of the main tools in the study of polygroups is the fundamental relations. These relations connect polygroups to groups, and on the other hand, introduce new important classes. So, we consider a crossed polymodule and by using the concept of fundamental relation, we obtain a crossed module. Moreover, we give a crossed polymodule morphism between them.Öğe Crossed polymodules and fundamental relations(Politechnica University of Bucharest, 2015) Alp, Murat; Davvaz, BijanIn this paper, we introduce the notion of crossed polymodule of poly-groups and we give some of its properties. Our results extend the classical results of crossed modules to crossed polymodules. One of the main tools in the study of polygroups is the fundamental relations. These relations connect polygroups to groups, and on the other hand, introduce new important classes. So, we consider a crossed polymodule and by using the concept of fundamental relation, we ob-tain a crossed module. Moreover, we give a crossed polymodule morphism between them.Öğe PELLANS SEQUENCE AND ITS DIOPHANTINE TRIPLES(PUBLICATIONS L INSTITUT MATHEMATIQUE MATEMATICKI, 2016) Irmak, Nurettin; Alp, MuratWe introduce a novel fourth order linear recurrence sequence {S-n} using the two periodic binary recurrence. We call it "pellans sequence" and then we solve the system ab + 1 = S-x, ac + 1 = S-y bc + 1 = S-z where a < b < c are positive integers. Therefore, we extend the order of recurrence sequence for this variant diophantine equations by means of pellans sequence.Öğe Pullback and pushout crossed polymodules(INDIAN ACAD SCIENCES, 2015) Alp, Murat; Davvaz, BijanIn this paper, we introduce the concept of pullback and pushout crossed polymodules and we describe the construction of pullback and pushout crossed polymodules. In particular, by using the notion of fundamental relation, we obtain a crossed module from a pullback crossed polymodule.Öğe Reduced diophantine quadruples with the binary recurrence G(n) = AG(n-1) - G(n-2)(OVIDIUS UNIV PRESS, 2015) Alp, Murat; Irmak, Nurettin; Szalay, LaszloGiven a positive integer A not equal 2. In this paper, we show that there do not exist two positive integer pairs {a, b} not equal {c, d} such that the values of ac + 1, ad + 1 and bc + 1, bd + 1 are the terms of the sequence {G(n)}(n >= 0) which satisfies the recurrence relation G(n) = AG(n-1) - G(n-2) with the initial values G(0) = 0, G(1) = 1.Öğe Some identities for generalized Fibonacci and Lucas sequences(Hacettepe University, 2013) Irmak, Nurettin; Alp, MuratIn this study, we define a generalization of Lucas sequence {pn}. Then we obtain Binet formula of sequence {pn}. Also, we investigate relationships between generalized Fibonacci and Lucas sequences. © 2013, Hacettepe University. All rights reserved.Öğe SOME IDENTITIES FOR GENERALIZED FIBONACCI AND LUCAS SEQUENCES(HACETTEPE UNIV, FAC SCI, 2013) Irmak, Nurettin; Alp, MuratIn this study, we define a generalization of Lucas sequence {p(n)}. Then we obtain Binet formula of sequence {p(n)}. Also, we investigate relationships between generalized Fibonacci and Lucas sequences.Öğe TRIBONACCI NUMBERS WITH INDICES IN ARITHMETIC PROGRESSION AND THEIR SUMS(UNIV MISKOLC INST MATH, 2013) Irmak, Nurettin; Alp, MuratIn this paper, we give a recurrence relation for the Tribonacci numbers with indices in aritmetics progression, {Trn+s} for 0 <= s < n We find sums of {T-rn} g for arbitrary integer r via matrix methods.
