Meta-Fibonacci Sequences, Binary Trees, and Extremal Compact Codes

A family of meta-Fibonacci sequences defined by variable order recursions

We define a family of meta-Fibonacci sequences where the order of the of recursion at stage n is a variable r(n), and the n^{th} term of a sequence is the sum of the previous r(n) terms. For the terms of any such sequence, we give upper and lower bounds which depend only on r(n).

Isoperimetric Sequences for Infinite Complete Binary Trees, Meta-Fibonacci Sequences and Signed Almost Binary Partitions

In this paper we demonstrate connections between three seemingly unrelated concepts. (1) The discrete isoperimetric problem in the infinite binary tree with all the leaves at the same level, $ {\mathcal T}_{\infty}$: The $n$-th edge isoperimetric number $δ(n)$ is defined to be $\min_{|S|=n, S \subset V({\mathcal T}_{\infty})} |(S,\bar{S})|$, where $(S,\bar{S})$ is the set of edges in the cut defined by $S$. (2) Signed almost binary partitions: This is the special case of the coin-changing problem where the coins are drawn from the set ${\pm (2^d - 1): $d$ is a positive integer}$. The quantity …

Spot-Based Generations for Meta-Fibonacci Sequences

For many meta-Fibonacci sequences it is possible to identify a partition of the sequence into successive intervals (sometimes called blocks) with the property that the sequence behaves "similarly" in each block. This partition provides insights into the sequence properties. To date, for any given sequence, only ad hoc methods have been available to identify this partition. We apply a new concept - the spot-based generation sequence - to derive a general methodology for identifying this partition for a large class of meta-Fibonacci sequences. This class includes the Conolly and Conway sequences…

On variants of Conway and Conolly's Meta-Fibonacci recursions

We study the recursions $A(n) = A(n-a-A^k(n-b)) + A(A^k(n-b))$ where $a \geq 0$, $b \geq 1$ are integers and the superscript $k$ denotes a $k$-fold composition, and also the recursion $C(n) = C(n-s-C(n-1)) + C(n-s-2-C(n-3))$ where $s \geq 0$ is an integer. We prove that under suitable initial conditions the sequences $A(n)$ and $C(n)$ will be defined for all positive integers, and be monotonic with their forward difference sequences consisting only of 0 and 1. We also show that the sequence generated by the recursion for $A(n)$ with parameters $(k,a,b) = (k,0,1)$, and initial conditions $A(1) …

The Meta-C-finite Ansatz

The Fibonacci numbers satisfy the famous recurrence $F_n = F_{n - 1} + F_{n - 2}$. The theory of C-finite sequences ensures that the Fibonacci numbers whose indices are divisible by $m$, namely $F_{mn}$, satisfy a similar recurrence for every positive integer $m$, and these recurrences have an explicit, uniform representation. We will show that $a(mn)$ has a uniform recurrence over $m$ for any C-finite sequence $a(n)$ and use this to automatically derive some famous summation identities.

金属立方体

Metallic cubes

本文研究推广斐波那契立方体和佩尔图的双参数图族，证明其为超立方体的诱导子图并给出规范分解，计算度量不变量和度分布，建立哈密顿性质，为潜在应用奠定基础。

Horadam 立方体

Horadam cubes

本文提出并研究了一个新的三参数图族 Horadam 立方体，其顶点数满足 Horadam 递推，并证明了它保留 Fibonacci 和 metallic 立方体的许多良好性质，同时分析了其分解、度量、计数性质及 Hamiltonian 路径与圈的存在性。

Extremal product-one free sequences and $|G|$-product-one free sequences of a metacyclic group

Let $G$ be a multiplicatively written finite group. We denote by $\mathsf E(G)$ the smallest integer $t$ such that every sequence of $t$ elements in $G$ contains a product-one subsequence of length $|G|$. In 1961, Erdős, Ginzburg and Ziv proved that $\mathsf E(G)\leq 2|G|-1$ for every finite abelian group $G$ and this result is known as the Erdős-Ginzburg-Ziv Theorem. In 2005, Zhuang and Gao conjectured that $\mathsf E(G)=\mathsf d(G)+|G|$ for every finite group, where $\mathsf d(G)$ is the small Davenport constant. Very recently, we confirmed this conjecture for the case when $G=\langle x, y|…

A Combinatorial interpretation of Hofstadter's G-sequence

We give a combinatorial interpretation of a classical meta-Fibonacci sequence defined by G(n) = n - G(G(n-1)) with the initial condition G(1) = 1, which appears in Hofstadter's 'Godel, Escher, Bach: An Eternal Golden Braid'. The interpretation is in terms of an infinite labelled tree. We then show a few corollaries about the behaviour of the sequence G(n) directly from the interpretation.

Erdős–Kleitman问题的极值族：缺失的构造

Extremal Families for the Erdős--Kleitman Problem: The Missing Constructions

本文针对Erdős–Kleitman问题，构造了1≤k≤m−1时的无穷多新极值族，验证了Frankl–Kupavskii元猜想，并对相关结果进行了定量扩展。

Nested Recurrence Relations With Conolly-Like Solutions

A nondecreasing sequence of positive integers is $(α,β)$-Conolly, or Conolly-like for short, if for every positive integer $m$ the number of times that $m$ occurs in the sequence is $α+ βr_m$, where $r_m$ is $1$ plus the 2-adic valuation of $m$. A recurrence relation is $(α, β)$-Conolly if it has an $(α, β)$-Conolly solution sequence. We discover that Conolly-like sequences often appear as solutions to nested (or meta-Fibonacci) recurrence relations of the form $A(n) = \sum_{i=1}^k A(n-s_i-\sum_{j=1}^{p_i} A(n-a_{ij}))$ with appropriate initial conditions. For any fixed integers $k$ and $p_1,p…

图族中具有奥尔条件的泛圈性

Pancyclicity in Graph Families with the Ore-Type Condition

本文在奥尔条件下证明了图族中每个顶点都包含长度为ℓ的泛圈，或图族为二分图，支持了Bondy的著名猜想，并证明了图族的泛圈性。