= Solution
Every discrete additive subgroup of $\mathbb R^2$ has the form $\mathbb Zv_1+\cdots+\mathbb Zv_r$ for linearly independent vectors, with $r=0,1,$ or $2$. A full <translation lattice> has $r=2$, and conversely every such integer span is a discrete cocompact subgroup. This classifies planar lattices up to a choice of basis.
A <frieze group> is a discrete group of planar <Euclidean isometries> whose <translation subgroup> is infinite cyclic and has finite index. Consider the <glide reflection> $g(x,y)=(x+1,-y)$. Its square is translation by $(2,0)$, whereas every odd power reverses orientation. Therefore $\langle g\rangle\cong\mathbb Z$, but none of its translation elements generates the whole group: they form the index-two subgroup $\langle g^2\rangle$.
To realize exactly this <frieze group>, repeat a small scalene triangular motif at the points $(n,(-1)^n)$, reflecting the motif when $n$ is odd. Choose the motif with three distinct side lengths and no internal symmetry. The displayed pattern has <glide reflection> symmetry $g$ and translation period $2$, but no pure horizontal reflection or vertical mirror. Distinct side lengths force any symmetry to preserve corresponding vertices, which rules out extra motif permutations; the sequence of heights then leaves exactly the powers of $g$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ii/paper-2-frieze.png]
{title=Alternating reflected scalene motifs with a glide-reflection generator and translation period two}
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