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A Fibonacci-Based G\"odel Numbering: $\Delta_0$ Semantics Without Exponentiation
Milan Rosko
https://arxiv.org/abs/2509.10382 https://arxiv.org/pdf/25…

A Fibonacci-Based Gödel Numbering: $Δ_0$ Semantics Without Exponentiation
We introduce a more efficient Gödel encoding scheme based on Zeckendorf representations of natural numbers, avoiding the use of unbounded exponentiation and without appealing to the fundamental theorem of arithmetic. The resulting encoding is injective, primitive recursive, and $Δ_0$-definable in weak arithmetical theories such as $IΔ_0 + Ω_1$. This allows for the construction of fixed points and the formalization of incompleteness theorems entirely within bounded arithmetic. Compared to tr…

Fibonacci sequences in $\mathbb F_p$
Menny Aka
https://arxiv.org/abs/2508.08016 https://arxiv.org/pdf/2508.08016 …

Fibonacci sequences in $\mathbb F_p$
We prove a conjecture due to Gica (Fibonacci Quarterly, 2008). Our proof is simple, uses only elementary abstract algebra, and generalizes to other recursive sequences.

I usually spend my Saturday faffing about not doing anything because I'm lazy but now I'm faffing about not doing anything because I can't because of a broken wrist. Definitely more frustrating.

Density Characterization with The Upper Bound of Density of Fibonacci Word
Duaa Abdullah, Jasem Hamoud
https://arxiv.org/abs/2509.00886 https://arxiv.org/p…

Density Characterization with The Upper Bound of Density of Fibonacci Word
This paper investigates the natural density and structural relationships within Fibonacci words, the density of a Fibonacci word is $\operatorname{DF}(F_k)=n/(n+m),$ where $m$ denote the number of zeros in a Fibonacci word and $n$ denote the units digit. Through analysis of these ratios and their convergence to powers of $φ$, we illustrate the intrinsic exponential growth rates characteristic of Fibonacci words. By considering the natural density concept for sets of positive integers, it is de…

Classical Fibonacci compositions
Brian Hopkins
https://arxiv.org/abs/2509.04493 https://arxiv.org/pdf/2509.04493 https://arxiv.org/html/2509.04493
arXiv:2509.04493v1 Announce Type: new
Abstract: There are three long-known types of restricted integer compositions whose counts match the Fibonacci sequence:\ one from ancient India and two from 19th century England. We give proofs of these enumeration results using tiling arguments and discuss how these can be used in combinatorial proofs of several Fibonacci number identities. With MacMahon's notion of conjugation, we show that, for every $n \ge 2$, the compositions of $n$ include subsets whose sizes satisfy the Fibonacci recurrence.
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Flux-driven charge and spin transport in a dimerized Hubbard ring with Fibonacci modulation
Souvik Roy, Soumya Ranjan Padhi, Tapan Mishra
https://arxiv.org/abs/2509.09542 https:…

Flux-driven charge and spin transport in a dimerized Hubbard ring with Fibonacci modulation
We study quantum transport in a one-dimensional Hubbard ring with dimerized nearest-neighbor hoppings and a Fibonacci-modulated onsite potential. For non-interacting case our analysis reveals that at half-filling, the charge current along with the Drude weight decreases with increasing onsite potential when inter-cell hopping dominates over the intra-cell hopping, while for dominating intra-cell hopping it shows non-monotonic behavior with sharp peak at certain critical modulation strength, ind…

Recursions, Trains, Trees, and Combinatorial Rod Set Algebra
Ethan D. Bolker, Debra K. Borkovitz, Katelyn Lee
https://arxiv.org/abs/2508.08392 https://arxi…

Recursions, Trains, Trees, and Combinatorial Rod Set Algebra
We explore a physical model of ordered sums of integers as trains of rods. The trains for a fixed, possibly infinite, set of rod lengths naturally correspond to nodes in a tree; relations among finite linear recursions encoded in the subtrees define algebraic operations on sets of rods. We use this algebra to prove classic identities for recursively defined sequences, to show that Lucas sequences are divisibility sequences, to characterize two-term linear Fibonacci identities, and to find the c…

Narayana numbers that are products of two Fibonacci numbers
Japhet Odjoumani
https://arxiv.org/abs/2508.02688 https://arxiv.org/pdf/2508.02688

Narayana numbers that are products of two Fibonacci numbers
Let $\{N_m\}_{m\ge0}$ be the Narayana's cows sequence given by $N_0=0$, $N_1=1=N_2=1$ and \[ N_{m+3}=N_{m+2}+N_m,\quad \text{ for }\; m\geq 0 \] and let $\{F_n\}_{n\ge0}$ be the Fibonacci sequence. In this paper we solve explicitely the Diophantine equation \[ N_m=F_nF_k, \] in positive unknowns $m,\,n$ and $k$. That is, we find the non-zero narayana numbers that are products of two Fibonacci numbers.

Some power series extensions of Fibonacci and Lucas polynomials
Johann Cigler
https://arxiv.org/abs/2509.04881 https://arxiv.org/pdf/2509.04881

Some power series extensions of Fibonacci and Lucas polynomials
We study formal power series which can be interpreted as interpolations of Fibonacci and Lucas polynomials with even (or odd) indices.

A Study of Fibonacci Cordial Labeling in Structured Graph Families
Sarbari Mitra, Soumya Bhoumik
https://arxiv.org/abs/2509.01823 https://arxiv.org/pdf/250…

A Study of Fibonacci Cordial Labeling in Structured Graph Families
A \emph{Fibonacci cordial labeling} of a graph \( G \) is an injective function \( f: V(G) \rightarrow \{F_0, F_1, \dots, F_n\} \), where \( F_i \) denotes the \( i^{\text{th}} \) Fibonacci number, such that the induced edge labeling \( f^*: E(G) \rightarrow \{0,1\} \), given by \( f^*(uv) = (f(u) + f(v)) \) $(\bmod\ 2)$, satisfies the balance condition \( |e_f(0) - e_f(1)| \le 1 \). Here, \( e_f(0) \) and \( e_f(1) \) represent the number of edges labeled 0 and 1, respectively. A gra…