We present our study of a set of solutions to the $SU(N)$ Yang-Mills equations of motion with fractional topological charge. The configurations are obtained numerically by minimizing the action with gradient flow techniques on a torus of size $l^2 \times(Nl)^2$ with twisted boundary conditions. We pay special attention to the large N limit, which is taken along a very peculiar sequence, with the number of colors N and the magnetic flux m selected respectively as the $n$-th and $n − 2$ terms of the Fibonacci sequence. We discuss the large N scaling of the solutions and analyze several gauge invariant quantities as the Polyakov loops. We also discuss the so-called Hamiltonian limit, with one of the large directions sent to infinity, where these instantons represent tunneling events between inequivalent pure gauge configurations.