This non-linear coupling resonance occurs when the spring (vertical) frequency is twice the frequency of a pendulum of the length at equilibrium. This is of interest because it is a very good mechanical analog to the v(r) = 2 v(z) resonance in the extraction region of a sector-focused cyclotron, where v(r) and v(z) are the radial and vertical betatron frequencies.
The mass required to be connected to a spring to induce this behavior can be determined as follows by noting the resonant condition:
v(mass on spring) = 2 v(pendulum)
sqrt[ k/m ] = 2 sqrt[ g/L ]
k = 4 mg/L
This means that you must add a weight so that the increase in length of the original spring is 1/3 of the original spring, or 1/4 of the length of the final spring (spring constant = mg/[L/4]).