Kombinatorna interpretacija verižnih razlomaka

master's thesis

• Author: Anamarija Zelić
• Mentor: Tomislav Pejković (mentor)
• Committee member: Tomislav Pejković (predsjednik povjerenstva)
• Committee member: Maja Starčević (član povjerenstva)
• Committee member: Vedran Čačić (član povjerenstva)
• Committee member: Matija Kazalicki (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-07-18, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Tema ovog diplomskog rada je jedna kombinatorna interpretacija verižnih razlomaka. U prvom poglavlju definirani su jednostavni verižni razlomci i opisana su neka njihova svojstva. Uvodi se ideja popločavanja \(n\times 1\) ploče domino i kvadratnim pločicama i pokazuje veza između takvih popločavanja i brojnika, odnosno nazivnika konvergenti jednostavnih verižnih razlomaka. Zatim su tvrdnje vezane uz verižne razlomke dokazane koristeći spomenutu kombinatornu interpretaciju, a u više slučajeva je usporedno dan i algebarski dokaz istih tvrdnji. Kombinatornom metodom obrađene su i neke familije verižnih razlomaka gdje se kao vrijednosti pojavljuju razlomci s Fibonaccijevim ili Lucasovim brojevima. U drugom poglavlju je opisana kombinatorna interpretacija složenih verižnih razlomaka te su dokazani analogoni nekih tvrdnji koje su pokazane za jednostavne verižne razlomke.

Abstract (english)

The topic of this graduation thesis is one combinatorial interpretation of continued fractions. In the first chapter simple continued fractions are defined and some of their properties are described. The idea of tiling \(n\times 1\) board by square and domino tiles is introduced and a connection between such tilings and numerators or denominators of convergents to simple continued fractions is established. Some results on continued fractions are proved using this combinatorial interpretation and in several cases algebraic proof of the same statements is given for comparison. Some families of continued fractions where values are rational numbers in which Fibonacci or Lucas numbers appear are also treated using the combinatorial method. In the second chapter the combinatorial interpretation is generalized to nonsimple continued fractions and some statements already proved in the case of simple continued fractions are proved for these more general continued fractions.

• Keywords: verižni razlomci; kombinatorna interpretacija; Fibonaccijevi brojevi; Lucasovi brojevi
• Keywords (english): continued fractions; combinatorial interpretation; Fibonacci numbers; Lucas numbers
• Language: croatian
• URN:NBN: urn:nbn:hr:217:693978
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2019-07-09 09:39:50