相关论文: Order and Chaos in Hofstadter's Q(n) Sequence

Hofstadter's Q-sequence remains an enigma fifty years after its introduction. Initially, the terms of the sequence increase monotonically by 0 or 1 at a time. But, Q(12)=8 while Q(11)=6, and monotonicity fails shortly thereafter. In this…

We continue work started in [1] concerning integer sequences q(n), n in N, defined by q(n) = q(n-q(n-1)) + f(n), with q(1) = 1. Here, f(n), with f(1) = 0, is a given sequence. We define F as the set of semi-infinite sequence f such that the…

The Hofstadter $H$ sequence is defined by $H(1) = 1$ and $H(n) = n-H(H(H(n-1)))$ for $n > 1$. If $\alpha$ is the real root of $x^3+x=1$ we show that the numbers $\alpha H(n) \mod 1$ are not uniformly distributed on $[0,1]$, but converge to…

I study the recurrence D(n)= D(D(n-1))+D(n-1-D(n-2)), D(1)=D(2)=1. Its definition has some similarity to that of Conway's sequence defined through a(n)= a(a(n-1))+a(n-a(n-1)), a(1)=a(2)=1. However, in contradistinction to the completely…

We investigate the conditions on an integer sequence f(n), n 2 N, with f(1) = 0, such that the sequence q(n), computed recursively via q(n) = q(n - q(n - 1)) + f(n), with q(1) = 1, exists. We prove that f(n + 1) - f(n) in {0,1}, n > 0, is a…

I propose and investigate the use of continuous functional equations for the study of meta-Fibonacci integer sequences. This exploratory study includes three sequences with quite different behavior: Conway's famous sequence $A(n)=…

The Hofstadter Q-sequence is a prominent example of nested recurrence. Despite decades of study, it is not even known whether Q(n) is defined for all n. Mantovanelli introduced a parity-perturbed variant $\widetilde{Q}$, obtained by adding…

The Hofstadter Q-sequence, with its simple definition, has defied all attempts at analyzing its behavior. Defined by a simple nested recurrence and an initial condition, the sequence looks approximately linear, though with a lot of noise.…

We study a perturbed variant of Hofstadter's $Q$-recursion \[ Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2))+(-1)^n, \qquad Q(1)=Q(2)=1 . \] Numerical experiments indicate that the sequence remains well defined for very large values of $n$ and exhibits an…

Hofstadter's G function is recursively defined via $G(0)=0$ and then $G(n)=n-G(G(n-1))$. Following Hofstadter, we vary the number $k$ of nested recursive calls in this equation and obtain a family of functions $(F\_k)$. Here we establish…

In this study, we explore the properties of certain solutions of Hofstadter's famous $V$-recurrence, defined by the nested recurrence relation $V(n)=V(n-V(n-1))+V(n-V(n-4))$. First, we discover the nature behind a finite chaotic…

Nested recurrence relations are highly sensitive to their initial conditions. The best-known nested recurrence, the Hofstadter $Q$-recurrence, generates sequences displaying a wide variety of behaviors. Most famous among these is the…

A set $A$ of positive integers is said to be Schreier if either $A = \emptyset$ or $\min A\ge |A|$. We give a bijective map to prove the recurrence of the sequence $(|\mathcal{K}_{n, p, q}|)_{n=1}^\infty$ (for fixed $p\ge 1$ and $q\ge 2$),…

We introduce a family of averaged meta-Fibonacci recursions $$ Q_{\alpha,m}(n) = 1+ \left\lfloor \alpha \frac1m \sum_{j=1}^m Q_{\alpha,m}(n-Q_{\alpha,m}(n-j)) \right\rfloor , $$ with initial conditions $$…

In this paper, we study the three-term nested recurrence relation $B(n)=B(n-B(n-1))+B(n-B(n-2))+B(n-B(n-3))$ subject to initial conditions where the first $N$ terms are the integers $1$ through $N$. This recurrence is the three-term analog…

Let $k\geq2$. Then the $k$-th order Fibonacci cube $\Gamma^{(k)}_{n}$ is the subgraph of the hypercube $Q_{n}$ induced by vertices without $k$ consecutive $1$s. The case $k=2$ corresponds to the classic Fibonacci cube $\Gamma_{n}$. There…

A random Fibonacci sequence is defined by the relation g_n = | g_{n-1} +/- g_{n-2} |, where the +/- sign is chosen by tossing a balanced coin for each n. We generalize these sequences to the case when the coin is unbalanced (denoting by p…

Hofstadter's $G$ function is recursively defined via $G(0)=0$ and then $G(n)=n-G(G(n-1))$. Following Hofstadter, a family $(F_k)$ of similar functions is obtained by varying the number $k$ of nested recursive calls in this equation. We…

The spectral statistic $\delta_n$ measures the fluctuations of the number of energy levels around its mean value. It has been shown that chaotic quantum systems display $1/f$ noise (pink noise) in the power spectrum $S(f)$ of the $\delta_n$…

Let $(a_n)_{n\ge 1}$ be the greedy self-generating sequence defined by $a_1=1$, $a_2=2$, and, for $k\ge 3$, by taking $a_k$ to be the least integer greater than $a_{k-1}$ that can be written as a sum of at least two consecutive earlier…