GCD properties in Hosoya's triangle (Q2894235)

The Hosoya sequence \(H(r, k)\) is defined by the recurrences \(H(r, k)=H(r-1, k)+H(r-2, k)\) and \(H(r, k)=H(r-1, k-1)+H(r-2, k-2)\) with certain initial coditions, (see \textit{H. Hosoya} [Fibonacci Q. 14, 173--179 (1976; Zbl 0359.10011)]). Hosoya proved that \(H(k, r)=F_{k}F_{r-k+1}\), where \(F_n\) is the Fibonacci sequence. The Hosoya sequence gives rise to the Hosoya triangle, analogous to the Pascal triangle.NEWLINENEWLINEThe present authors prove GCD properties for numbers in the Star of David and other polygon configurations formed in the Hosoya triangle. They also give a criterion to determine whether a sequence of points in a polygon or in a rhombus configurations in the Hosoya triangle have GCD equal to one. Several known results from the Pascal triangle are transferred to the Hosoya triangle.