[math-fun] Type 3-1-3 babbling brook oscillators?

I was looking at simple formal structures for "babbling brook" type Life oscillators. If we had a babbling brook of length 3, where, when the brook was all dead, each end cell had 3 live neighbors and the middle cell had 1, we could see the following p4 oscillation pattern: 000, 101, 111, 010, and repeat. If such an oscillator exists, I can't find it. Is it possible to stabilize the bed of a type 3-1-3 oscillator?