Further results on the enumeration of hamilton paths in Cayley digraphs on semidirect products of cyclic groups

First, let m and n be positive integers such that n   is odd and gcd(m,n)=1gcd(m,n)=1. Let G   be the semidirect product of cyclic groups given by G=Z8m⋊Z2n=〈x,y:x8m=1,y2n=1,andyxy-1=x4m+1〉. Then the number of hamilton paths in Cay(G:x,y)Cay(G:x,y) (with initial vertex 1) is one fewer than the number of visible lattice points that lie on the closed quadrilateral whose vertices in consecutive order are (0,0)(0,0), (4mn2+2n,16m2n)(4mn2+2n,16m2n), (n,4m)(n,4m), and (0,8m)(0,8m). Second, let m and n be positive integers such that n is odd. Let G   be the semidirect product of cyclic groups given by G=Z4m⋊Z2n=〈x,y:x4m=1,y2n=1,andyxy-1=x2m-1〉. Then the number of hamilton paths in Cay(G:x,y)Cay(G:x,y) (with initial vertex 1) is (3m-1)n+m⌊(n+1)/3⌋+1(3m-1)n+m⌊(n+1)/3⌋+1.