Sequences involving square zig-zag shapes

TL;DR

This paper introduces square k-zig-zag shapes on a grid, analyzes their properties as digraphs, and derives integer sequences and recurrence relations based on shortest paths from a base vertex.

Contribution

It defines a new geometric shape and explores its combinatorial properties, including the derivation of recurrence relations for associated integer sequences.

Findings

Integer sequences from k-zig-zag shapes are characterized

Higher-order recurrence relations are established

Method for calculating shortest paths in these shapes

Abstract

We define a so-called square $k$-zig-zag shape as a part of the regular square grid. Considering the shape as a $k$-zig-zag digraph, we give values of its vertices according to the number of the shortest paths from a base vertex. It provides several integer sequences, whose higher-order homogeneous recurrences are determined by the help of a special matrix recurrence.