Betti numbers of powers of path ideals of cycles

Let Jn,m = (x₁⋯ xₘ,x₂ ⋯ xₘ₊₁,…,xₙx₁⋯ xₘ₋₁) be the m-path ideal of a cycle of length n ≥ 5 over a polynomial ring S = k[x₁,…,xₙ]. Let t≥ 1 be an integer. We show that Jn,mᵗ has a linear free resolution and give a precise formula for all of its Betti numbers when $m = n-1, n-2$.