We present an analysis of AAVSO visual observations of the Mira variable R Cen from 1918 to 2000. The period of the dominant mode has been steadily decreasing from 550 days at JD 2,434,000 (1951) to its present value of 505-510 days. In the same interval, the pulsational amplitude has decreased by 3 mag, from 5.5-11.8 V to 6.3-9.1 V. We suggest that both are caused by a He-shell flash, as the period decrease is similar to that of other He-shell flash stars such as R Hya, R Aql, and T UMi. The period change is consistent with the luminosity drop expected immediately after the flash, as predicted by He-shell flash models for stars of 2-3 Msolar or less. The light curve shows the familiar pattern of alternating deep and shallow minima, giving the appearance of double maxima. While the amplitude of the main mode has decreased 3 mag in the last 50 years, the amplitude of the secondary mode near 274 days has remained almost constant, so that the double maxima have nearly vanished from the light curve in recent years. The power spectrum between 1930 and 1966 shows harmonics up to 8 times the main frequency at 1/548 cycle day-1. The most likely explanation for the double-peaked light curve is a resonance between two modes.