Answer
Yes, there is a relationship among the wavelengths.
Work Step by Step
Energy is conserved. The sum of the two photon energies equals the single photon's energy.
$$E_{1} + E_{2} = E_{3}$$
For photons, E = hf, where E is the energy, h is Planck's constant, and f is the photon's vibrational frequency.
Wavelength is inversely proportional to frequency. That is, the speed of light is frequency multiplied by wavelength.
$$c = f \lambda$$
Solve for $\lambda = \frac{c}{f} $.
Finally, we are ready to relate the 3 wavelengths.
$$E_{1} + E_{2} = E_{3}$$
$$h(\frac{c}{\lambda_{1}} +\frac{c}{\lambda_{2}}) = h \frac{c}{\lambda_{3}} $$
$$ \frac{1}{\lambda_{1}} +\frac{1}{\lambda_{2}} = \frac{1}{\lambda_{3}} $$
So the sum of the reciprocals of the two photon wavelengths, emitted when the electron transitions from 4 to 3 and then from 3 to 1, is equal to the reciprocal of the wavelength of light emitted when the electron transitions from quantum level 4 straight down to quantum level 1.
