Zeckendorf's Theorem implies that the Fibonacci number Fn is the smallest positive integer that cannot be written as a sum of non-consecutive previous Fibonacci numbers. Catral et al. studied a variation of the Fibonacci sequence, the Fibonacci Quilt sequence: the plane is tiled using the Fibonacci spiral, and integers are assigned to the squares of the spiral such that each square contains the smallest positive integer that cannot be expressed as the sum of non-adjacent previous terms. This adjacency is essentially captured in the differences of the indices of each square: the i-th and j-th squares are adjacent if and only if ∣i−j∣∈{1,3,4} or {i,j}={1,3}. We consider a generalization of this construction: given a set of positive integers S, the S-legal index difference (S-LID) sequence (an)n=1∞ is defined by letting an to be the smallest positive integer that cannot be written as ∑ℓ∈Laℓ for some set L⊂[n−1] with ∣i−j∣∈/S for all i,j∈L. We discuss our results governing the growth of S-LID sequences, as well as results proving that many families of sets S yield S-LID sequences which follow simple recurrence relations.
@article{arxiv.2210.10577,
title = {Recurrence Relations for $S$-Legal Index Difference Sequences},
author = {Guilherme Zeus Dantas e Moura and Andrew Keisling and Astrid Lilly and Annika Mauro and Steven J. Miller and Matthew Phang and Santiago Velazquez Iannuzzelli},
journal= {arXiv preprint arXiv:2210.10577},
year = {2023}
}