Tilings of a Honeycomb Strip and Higher Order Fibonacci Numbers

2023

In general, in the midst of History of Mathematics textbooks, we are faced with a discussion due to curiosity about the emblematic Fibonacci Sequence, whose popularization occurred with the proposition of the reproduction model of immortal rabbits. On the other hand, in the comparison of the multiple approaches and discussions of certain subjects in Elementary Mathematics, in the present work, we highlight combinatorial interpretations that, with the support of a characteristic and fundamental reasoning for the mathematics teacher, can be generalized and formalize some eminently intuitive components. In particular, this work deals with properties derived from the notion of tiling and decomposition of an integer that, depending on the board, will correspond to the numbers of the Fibonacci Sequence. We bring a theoretical discussion supported by great names that research in this area.

Cornell University - arXiv, 2018

In two papers, Little and Sellers introduced an exciting new combinatorial method for proving partition identities which is not directly bijective. Instead, they consider various sets of weighted tilings of a 1 × ∞ board with squares and dominoes, and for each type of tiling they construct a generating function in two different ways, which generates a q-series identity. Using this method, they recover quite a few classical q-series identities, but Euler's Pentagonal Number Theorem is not among them. In this paper, we introduce a key parameter when constructing the generating functions of various sets of tilings which allows us to recover Euler's Pentagonal Number Theorem along with an infinite family of generalizations.

International Journal of Mathematical Education in Science and Technology, 2008

Publicat ion det ails, including inst ruct ions f or aut hors and subscript ion inf ormat ion: ht t p: / / www. t andf online. com/ loi/ t mes20 This note shows a combinatorial approach to some identities for generalized Fibonacci numbers. While it is a straightforward task to prove these identities with induction, and also by arithmetical manipulations such as rearrangements, the approach used here is quite simple to follow and eventually reduces the proof to a counting problem.

We give a combinatorial interpretation, an explicit formula and some other properties of hyperfibonacci numbers. Further, we deduce relationships between Fibonacci, hyperfibonacci, and incomplete Fibonacci numbers.

2021

Axioms

There is ongoing research into combinatorial methods and approaches for linear and recurrent sequences. Using the notion of a board defined for the Fibonacci sequence, this work introduces the Padovan sequence combinatorial approach. Thus, mathematical theorems are introduced that refer to the study of the Padovan combinatorial model and some of its extensions, namely Tridovan, Tetradovan and its generalization (Z-dovan). Finally, we obtained a generalization of the Padovan combinatorial model, which was the main result of this research.

Carpathian Mathematical Publications, 2020

In this paper, we find new relations involving the Pell-Padovan sequence which arise as determinants of certain families of Toeplitz-Hessenberg matrices. These determinant formulas may be rewritten as identities involving sums of products of Pell-Padovan numbers and multinomial coefficients. In particular, we establish four connection formulas between the Pell-Padovan and the Fibonacci sequences via Toeplitz-Hessenberg determinants.

International Journal of Mathematical Education in Science and Technology, 2008

Fibonacci Quarterly, 2004

We study generalizations of two well-known statistics on linear square-and-domino tilings by considering only those dominos the right half of which covers a multiple of k, where k is a fixed positive integer. Using the method of generating functions, we derive explicit expressions for the joint distribution polynomials of the two statistics with the statistic that records the number of squares in a tiling. In this way, we obtain two families of q-generalizations of the Fibonacci polynomials. When k=1, our formulas reduce to known results concerning previous statistics. Special attention is paid to the case k=2. As a byproduct of our analysis several combinatorial identities are obtained.