Related papers: A Slow Relative of Hofstadter's Q-Sequence

The Fibonacci sequence has been generalized in many ways. One of them is defined by the relation $t_n=at_{n-1}+t_{n-2}$ if $n$ is even, $t_n=bt_{n-1}+t_{n-2}$ if $n$ is odd, with initial values $t_0=0$ and $t_1=1$, where $a$ and $b$ are…

The famous (3n + 1) or Collatz conjecture has admitted some progress over the last several decades towards the conclusion that the conjecture is true (i.e. that all Collatz sequences will eventually reach a value of one), but has stubbornly…

Let (F_n^{(k)})_{n\geq -(k-2)} be the k-generalized Fibonacci sequence, defined as the linear recurrence sequence whose first k terms are \(0, 0, \ldots, 0, 1\), and whose subsequent terms are determined by the sum of the preceding k terms.…

We prove some new results and unify the proofs of old ones involving complete monotonicity of expressions involving gamma and $q$-gamma functions, $0 < q < 1$. Each of these results implies the infinite divisibility of a related probability…

We introduce a new permutation statistic, namely, the number of cycles of length $q$ consisting of consecutive integers, and consider the distribution of this statistic among the permutations of $\{1,2,...,n\}$. We determine explicit…

The sequence starts with a(1) = 1; to extend it one writes the sequence so far as XY^k, where X and Y are strings of integers, Y is nonempty and k is as large as possible: then the next term is k. The sequence begins 1, 1, 2, 1, 1, 2, 2, 2,…

Labeled infinite trees provide combinatorial interpretations for many integer sequences generated by nested recurrence relations. Typically, such sequences are monotone increasing. Several of these sequences also have straightforward…

Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by the recurrence $F_{n+2}=F_{n+1}+F_n$, for $n\geq 0$, where $F_0=0$ and $F_1=1$. There are several generalizations of this sequence and also several interesting identities. In this…

We show that essentially the Fibonacci sequence is the unique binary recurrence which contains infinitely many three-term arithmetic progressions. A criterion for general linear recurrences having infinitely many three-term arithmetic…

A constant term sequence is a sequence of rational numbers whose $n$-th term is the constant term of $P^n(\boldsymbol{x}) Q(\boldsymbol{x})$, where $P(\boldsymbol{x})$ and $Q(\boldsymbol{x})$ are multivariate Laurent polynomials. While the…

Non-linear recurrences which generate integers in a surprising way have been studied by many people. Typically people study recurrences that are linear in the highest order term. In this paper I consider what happens when the recurrence is…

Let us call a sequence of numbers heapable if they can be sequentially inserted to form a binary tree with the heap property, where each insertion subsequent to the first occurs at a leaf of the tree, i.e. below a previously placed number.…

We study the binary Ehrenfeucht Mycielski sequence seeking a balance between the number of occurrences of different binary strings. There have been numerous attempts to prove the balance conjecture of the sequence, which roughly states that…

We study the parity-perturbed Hofstadter-type recursion $$ Q(1)=Q(2)=1,\qquad Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2))+(-1)^n . $$ We prove, by computer-certified finite-state induction, that this recursion is well-defined for all $(n\ge 1)$. The proof…

We explore a family of nested recurrence relations with arbitrary levels of nesting, which have an interpretation in terms of fixed points of morphisms over a countably infinite alphabet. Recurrences in this family are related to a number…

In this article, we reduce the unsolved problem of convergence of Collatz sequences to convergence of Collatz sequences of odd numbers that are divisible by 3. We give an elementary proof of the fact that a Collatz sequence does not…

Denote by p_k the k-th power sum symmetric polynomial n variables. The interpretation of the q-analogue of the binomial coefficient as Hilbert function leads us to discover that n consecutive power sums in n variables form a regular…

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…

In this work, we investigate $H$-recurrence which is defined by $H(n) = H(n-H(n-2)) + H(n-H(n-3))$ thanks to a recent approach to certain recurrences such as Conway and Hofstadter $Q$ recursions.

The Collatz conjecture, which posits that any positive integer will eventually reach 1 through a specific iterative process, is a classic unsolved problem in mathematics. This research focuses on designing an efficient algorithm to compute…