In 2003, Benoit Cloitre entered a family of sequences in the OEIS that we call hiccup sequences. We collect the various claims, observations, and proofs of properties of these sequences that have been entered in the OEIS over the years, and present a unified approach, inspired by a remarkable theorem of Bosma, Dekking, and Steiner.

We extend the work of Kimberling and Moses, Zaslavsky, and Bosma et al. on anti-recurrence sequences. Kimberling and Moses formulated several questions about these sequences, which together suggest the meta-conjecture that every anti-recurrence sequence is the sum of a linear progression and an automatic sequence. We solve this conjecture under a restriction on the linear form that generates the anti-recurrence.

We use the automatic theorem prover Walnut to resolve various open problems from the OEIS (On-Line Encyclopedia of Integer Sequences) and beyond. Specifically, we clarify the structure of sequence A260311, which concerns runs of sums of upper Wythoff numbers. We extend a result of Hajdu, Tijdeman, and Varga on polynomials with nonzero coefficients modulo a prime. Additionally, we settle open problems related to the anti-recurrence sequences A265389 and A299409, as well as the sumfree sequences A026471 and A026475. Our findings also give rise to new open problems.

We prove that for a suitably nice class of random substitutions, their corresponding subshifts have automorphism groups that contain an infinite simple subgroup and a copy of the automorphism group of a full shift. Hence, they are countable, non-amenable and non-residually finite. To show this, we introduce the concept of shuffles and generalised shuffles for random substitutions, as well as a local version of recognisability for random substitutions that will be of independent interest. Without recognisability, we need a more refined notion of recognisable words in order to understand their automorphisms. We show that the existence of a single recognisable word is often enough to embed the automorphism group of a full shift in the automorphism group of the random substitution subshift.

We extend the notion of boycotts between players in cooperative games to boycotts between coalitions. We prove that convex games offer a proper setting for studying these games. Boycotts have a heterogeneous effect. Individual players that are targeted by many-on-one boycotts suffer most, while non-participating players may actually benefit from a boycott.

We want to find the convex combination S of iid Bernoulli random variables that maximizes P(S ≥ t) for a given threshold t. Endre Csóka conjectured that such an S is an average if t ≥ p, where p is the success probability of the Bernoulli random variables. We prove this conjecture for a range of p and t

Wythoff Nim is a classical combinatorial game of queen moves on a chessboard. There are many ways to describe its P-positions (safe positions to move to). One way is to code them by the Fibonacci word 010010100100101.., which is the unique fixed point of the substitution of 0 by 01, and of 1 by 0. The coordinates of the n-th P-position are encoded by the location of the n-th zero and the n-th one in the Fibonacci word. We show that a minor modification of the rules of Wythoff Nim leads to a game with P-positions that are coded by 010010010010100100.. This word can be derived by deleting all 2’s from the Tribonacci word, which is the unique fixed point of the substitution of 0 by 01, of 1 by 02, and of 2 by 0.

Let S and X be independent random variables, assuming values in the set of non-negative integers, and suppose further that both E(S) and E(X) are integers satisfying E(S) ≥ E(X). We establish a sufficient condition for the tail probability P(S ≥ E(S)) to be larger than the tail P(S + X ≥ E(S + X)), when the mean of S is equal to the mode.

This paper repurposes the classic insight from network theory that long-distance connections drive disease propagation into a strategy for controlling a second wave of Covid-19. We simulate a scenario in which a lockdown is first imposed on a population and then partly lifted while long-range transmission is kept at a minimum. Simulated spreading patterns resemble contemporary distributions of Covid- 19 across EU member states, German and Italian regions, and through New York City, providing some model validation. Results suggest that our proposed strategy may significantly reduce peak infection. We also find that post-lockdown flare-ups remain local longer, aiding geographical containment. These results suggest a tailored policy in which individuals who frequently travel to places where they interact with many people are offered greater protection, tracked more closely, and are regularly tested. This policy can be communicated to the general public as a simple and reasonable principle: Stay nearby or get checked.

Since the 1960s China and India have engaged in a dispute about the demarcation of their shared border. This territorial dispute led to a brief war in 1962, and recurring flare-ups over the following decades, including during the summer of 2020. The potential for further escalation of this dispute poses significant risks to Indian and Chinese civilians, US foreign policy objectives, and the stability of the international economic system. Despite the importance of this dispute, there have been relatively few attempts to understand the correlates of Chinese incursions. This paper addresses this important question by leveraging past work on the study of conflicts between states to derive a set of testable explanations about the impact of China–India relations, internal political affairs, international political issues, and domestic economic factors on the likelihood of incursions. The study uses 15 years of original data on monthly Chinese incursions into India along with a monthly dataset containing 18 independent variables, to develop a detailed statistical understanding of the factors that trigger Chinese incursions across the Indian border with a lead time between 1 and 6 months. The quantitative study finds that Chinese incursions are more likely when Chinese leadership is early in their tenure, but more likely when Indian leadership is in the later stages of their tenure. The results also show that closer cooperation between India and the US may trigger additional Chinese incursions into India. Finally, lower consumer confidence in the Chinese economy is consistently related to an increased likelihood of incursions. These findings have implications for the maintenance of peace and India’s national security policies. Periods of Chinese uncertainty, particularly when their economy exhibits weakness and when Chinese leaders are in the early stages of their tenure are more likely to experience incursions. Further, the strengthening of the US–Indian alliance, as well as increased conflict between India and Pakistan, create the potential for an elevated risk of incursions. During these periods India should likely be on higher alert, while India and Indian allies should signal the importance of diplomatic solutions for the dispute.

Suppose that some objects are hidden in a finite set S of hiding places that must be examined one by one. The cost of searching subsets of S is given by a submodular function, and the probability that all objects are contained in a subset is given by a supermodular function. We seek an ordering of S that finds all the objects with minimal expected cost. This problem is NP-hard, and we give an efficient combinatorial 2-approximation algorithm, generalizing analogous results in scheduling theory. We also give a new scheduling application where a set of jobs must be ordered subject to precedence constraints to minimize the weighted sum of some concave function of the completion times of subsets of jobs. We go on to give better approximations for submodular functions with low total curvature, and we give a full solution when the problem is what we call series-parallel decomposable. Next, we consider a zero-sum game between a cost-maximizing hider and a cost-minimizing searcher. We prove that the equilibrium mixed strategies for the hider are in the base polyhedron of the cost function, suitably scaled, and we solve the game in the series-parallel decomposable case, giving approximately optimal strategies in other cases.

We study search games in which the hider may hide in a finite number of locations. We assume that the cost of searching these locations does not depend on the order in which the locations are searched. From these assumptions we derive that the cost function is submodular, thus placing search games with an immobile hider in the context of coalitional games.

We give an explicit evaluation, in terms of products of Jacobsthal numbers, of the Hankel determinants of order a power of two for the period-doubling sequence. We also explicitly give the eigenvalues and eigenvectors of the corresponding Hankel matrices. Similar considerations give the Hankel determinants for other orders.