Related papers: A note on closed-form representation of Fibonacci …
We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already,…
In this paper we determine some properties of Fibonacci octonions. Also, we introduce the generalized Fibonacci-Lucas octonions and we investigate some properties of these elements.
We present a different combinatorial interpretations of Lucas and Gibonacci numbers. Using these interpretations we prove several new identities, and simplify the proofs of several known identities. Some open problems are discussed towards…
In this paper, we consider several combinatorial problems whose enumeration leads to the odd-indexed Fibonacci numbers, including certain types of Dyck paths, block fountains, directed column-convex polyominoes, and set partitions with no…
Using the classic two's complement notation of signed integers, the fundamental arithmetic operations of addition, subtraction, and multiplication are identical to those for unsigned binary numbers. We introduce a Fibonacci-equivalent of…
In this article we give two different ways of representations of circular words. Representations with tuples are intended as a compact notation, while representations with trees give a way to easily process all conjugates of a word. The…
By investigating a recurrence relation about functions, we first give alternative proofs of various identities on Fibonacci numbers and Lucas numbers, and then, make certain well known identities visible via certain trivalent graph…
The sequence $F_{dn+h}$ and its convolutions have (for $h=0$) been studied in a recent paper at the arxiv [arXiv:2603.08636]. The instance with general $h$ is more involved and uses Chebyshev polynomials.
This paper presents a new algorithm for the fast, shared memory, multi-core computation of augmented contour trees on triangulations. In contrast to most existing parallel algorithms our technique computes augmented trees, enabling the full…
We give a representation for labeled ordered trees that supports labeled queries such as finding the i-th ancestor of a node with a given label. Our representation is succinct, namely the redundancy is small-o of the optimal space for…
In this paper, we present a new method for determining the optimal pebbling number of a complete binary tree. This method reveals a curious connection between the optimal pebbling numbers of complete binary trees and the Conolly-Fox…
We introduce a representation via (n+1)-colored graphs of compact n-manifolds with (possibly empty) boundary, which appears to be very convenient for computer aided study and tabulation. Our construction is ageneralization to arbitrary…
We introduce a new variation of Tree Encoding with Nested Intervals, find connections with Materialized Path, and suggest a method for moving parts of the hierarchy.
Similar to topological spaces, we introduce the Cantor-Bendixson rank of a tree $T$ by repeatedly removing the leaves and the isolated vertices of $T$ using transfinite recursion. Then, we give a representation of a tree $T$ as a leafless…
Trees or rooted trees have been generously studied in the literature. A forest is a set of trees or rooted trees. Here we give recurrence relations between the number of some kind of rooted forest with $k$ roots and that with $k+1$ roots on…
We use combinatorial and generating function techniques to enumerate various sets of involutions which avoid 231 or contain 231 exactly once. Interestingly, many of these enumerations can be given in terms of $k$-generalized Fibonacci…
We develop new closed form representations of sums of (n + {\alpha})th shifted harmonic numbers and reciprocal binomial coefficients in terms of {\alpha}th shifted harmonic numbers. Some interesting new consequences and illustrative…
We describe a simple method that produces automatically closed forms for the coefficients of continued fractions expansions of a large number of special functions. The function is specified by a non-linear differential equation and initial…
We define the notion of circular words, then consider on such words a constraint derived from the Fibonacci condition. We give several results on the structure of these circular words, then mention possible applications to various…
This note contains a report of a proof by computer that the Fibonacci group F(2,9) is automatic. The automatic structure can be used to solve the word problem in the group. Furthermore, it can be seen directly from the word-acceptor that…