Nathaniel D. Emerson
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).
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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) …
Recursions Satisfied by Families of Determinants with Applications to Resistance Distance
本文提出一种通过迭代拉普拉斯展开得到行列式族递推关系的六步半自动算法，可计算图的电阻距离闭式公式，证明了线性3-树的渐近猜想系数，同时提供相关入门综述以吸引斐波那契数研究者参与该领域。
In a recent paper, Frank Ruskey asked whether every linear recurrent sequence can occur in some solution of a meta-Fibonacci sequence. In this paper, we answer his question in the affirmative for recurrences with positive coefficients.
In this article we obtain an explicit formula in terms of the partitions of the positive integer $n$ to express the $n$-th term of a wide class of sequences of numbers defined by recursion. Our proof is based only on arithmetics. We compare our result with similar formulas obtained with different approaches already in the XIX century. Examples are given for Bernoulli, Euler and Fibonacci numbers.
In this note we investigate the solutions of certain meta-Fibonacci recurrences of the form $f(n)=f(n-f(n-1))+f(n-2)$ for various sets of initial conditions. In the case when $f(n)=1$ for $n\leq 1$, we prove that the resulting integer sequence is closely related to the function counting binary partitions of a certain type.
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…
An Introductory Survey of Recursions in the Computation of Resistance Distance
本文是一篇电阻距离计算中递归方法的入门综述，介绍了电阻距离的定义、应用领域、计算方法，提及斐波那契等序列常出现在相关研究中，旨在唤起《斐波那契季刊》读者对该领域的兴趣。
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 sequence, a type of meta-Fibonacci sequence.
We look at a family of meta-Fibonacci sequences which arise in studying the number of leaves at the largest level in certain infinite sequences of binary trees, restricted compositions of an integer, and binary compact codes. For this family of meta-Fibonacci sequences and two families of related sequences we derive ordinary generating functions and recurrence relations. Included in these families of sequences are several well-known sequences in the Online Encyclopedia of Integer Sequences (OEIS).
We present a natural, combinatorial problem whose solution is given by the meta-Fibonacci recurrence relation $a(n) = \sum_{i=1}^p a(n-i+1 - a(n-i))$, where $p$ is prime. This combinatorial problem is less general than those given in [3] (B. Jackson, F. Ruskey, 2006) and [4] (F. Ruskey, C. Deugau, 2009), but it has the advantage of having a simpler statement.
Recursions, Trains, Trees, and Combinatorial Rod Set Algebra
通过杆的列车物理模型，建立树结构与递归序列的代数对应，证明经典恒等式、Lucas序列整除性、二项Fibonacci恒等式及Borwein三项式分圆因子。
The Fibonacci numbers are the prototypical example of a recursive sequence, but grow too quickly to enumerate sets of integer partitions. The same is true for the other classical sequences $a(n)$ defined by Fibonacci-like recursions: the tribonacci, Padovan, Pell, Narayana's cows, and Lucas sequences. For each sequence $a(n)$, however, we can define a related sequence $\textrm{sa}(n)$ by defining $\textrm{sa}(n)$ to have the same recurrence and initial conditions as $a(n)$, except that $\textrm{sa}(2n)=\textrm{sa}(n)$. Growth is no longer a problem: for each $n$ we construct recursively a set …
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 …
Recursions and characteristic polynomials of the Rows of the Circuit Array
本文研究 Circuit Array 各行的多项式递归，证明其单变量与多变量表述对应特征多项式的根仅为 9 的幂，并通过零化子方法验证初始情形及主要子情形。
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.
This paper further generalizes a recent result of Shannon and Ollerton who resurrected an old identity due to Ledin. This paper generalizes the Ledin-Shannon-Ollerton result to all the metallic sequences. The results give closed formulas for the sum of products of powers of the first $n$ integers with the first $n$ members of the metallic sequence. Three key innovations of this paper are i) reducing the proof of the generalization to the solution of a system of 4 simultaneous recursions; ii) use of the shift operation to prove equality of polynomials; and iii) new OEIS sequences arising from t…
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。
Rogers-Ramanujan exact sequences and free representations over free generalized vertex algebras
本研究从自由广义顶点代数的自由表示视角研究Rogers-Ramanujan递推，构造相关短正合序列推广已有结果，揭示其与共变空间的关联，还实现广义递推及相关模的正合序列表示，并应用于特征分解公式研究。
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。
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…
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.
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