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Kombinatorični dokazi Fibonaccijevih in sorodnih identitet
ID Vidmar, Katarina (Author), ID Kuzman, Boštjan (Mentor) More about this mentor... This link opens in a new window

URLURL - Presentation file, Visit http://pefprints.pef.uni-lj.si/4725/ This link opens in a new window

Abstract
V diplomskem delu obravnavamo kombinatorične dokaze Fibonaccijevih in sorodnih identitet. Kombinatorični dokazi se izkažejo za močno orodje, ki je uporabno na različnih matematičnih področjih, kot so teorija grup, teorija množic, analiza ter teorija grafov. Pred samim dokazovanjem identitet, ki vsebujejo Fibonaccijeva ali sorodna števila, si pogledamo različne vrste dokazov, kot so algebrski direktni dokaz, dokaz z indukcijo, dokaz s protislovjem ter vizualni dokaz in jih ilustriramo na primerih. Nato opredelimo Fibonaccijeva števila ter vpeljemo njihovo kombinatorično interpretacijo. Obravnavamo tudi posplošitve Fibonaccijevih števil. Ogledamo si zaokrožen izbor Fibonaccijevih in sorodnih identitet, ki jih kombinatorično dokažemo predvsem z metodo dvojnega štetja. V diplomsko delo je vključenih tudi nekaj identitet, ki so bile na kombinatoričen način dokazane nedavno, naveden pa je tudi kombinatorični dokaz Binetove formule. Nazadnje nekaj konceptov ilustriramo tudi s pomočjo Geogebre.

Language:Slovenian
Keywords:kombinatorični dokaz
Work type:Bachelor thesis/paper
Typology:2.11 - Undergraduate Thesis
Organization:PEF - Faculty of Education
Year:2017
PID:20.500.12556/RUL-95889 This link opens in a new window
COBISS.SI-ID:11727177 This link opens in a new window
Publication date in RUL:25.09.2017
Views:2084
Downloads:272
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Secondary language

Language:English
Title:Combinatorial proofs of Fibonacci and related identities
Abstract:
In this diploma thesis, the combinatorial proofs of Fibonacci and related identities are discussed. The combinatorial proofs prove to be strong tools, useful in different mathematical fields, such as group theory, set theory, analysis and theory of graphs. Before the proving of identities containing Fibonacci or related numbers, different types of proofs, such as direct algebraic proof, proof with mathematical induction, proof by contradiction and visual proof, are examined and illustrated with cases. Then, Fibonacci numbers are defined and their combinatorial interpretation introduced. The generalisations of Fibonacci numbers are also discussed. A rounded selection of Fibonacci numbers and similar identities are reviewed, and combinatorically proven, in particular with the double counting method. The diploma thesis also discusses some identities that have only recently been combinatorically proven; a combinatorial proof for the Binet’s formula is also cited. Finally, some of the concepts are illustrated also with the help of GeoGebra.

Keywords:combinatorial proof

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