In this article, dedicated with admiration and friendship to Chaos and Difference (and hence Recurrence) Equations Guru Saber Elaydi, I give a new approach and a new algorithm for Chomp, David Gale’s celebrated combinatorial game. This work is inspired by Xinyu Sun’s “ultimate-periodicity” conjecture and by its brilliant proof by high-school-student Steven Byrnes. The algorithm is implemented i...

We describe algorithm Hyper which can be used to nd all hypergeometric solutions of linear recurrences with polynomial coeecients.

In the last few readings we dealt with program correctness. We discussed correctness of nonrecursive as well as of recursive algorithms. In both cases, the proofs of correctness went by induction. As part of a proof of correctness, we show that the algorithm terminates. For example, we prove that a certain loop has to terminate after some number of iterations, or that the longest chain of recur...

Let {an}^_ao be a sequence which satisfies a linear recurrence of order k +1. We are herein concerned with the lacunary subsequences {®mn+b}Z=-ao> where m and b are fixed integers, so called because they consist of the terms from {aJ with lacunae, or gaps, of length m between them. In [5], [2], and [3] it has been shown that, for any m and A, the subsequences {amn+b} also satisfy a linear recur...

A binomial identity ((1) below), which relates the famous Apéry numbers and the sums of cubes of binomial coefficients (for which Franel has established a recurrence relation almost 100 years ago), can be seen as a particular instance of a Legendre transform between sequences. A proof of this identity can be based on the more general fact that the Apéry and Franel recurrence relations themselve...

In this paper, we present an approach to recover the dynamics from recurrences of a system and then generate (multivariate) twin surrogate (TS) trajectories. In contrast to other approaches, such as the linear-like surrogates, this technique produces surrogates which correspond to an independent copy of the underlying system, i.e. they induce a trajectory of the underlying system visiting the a...