November Prog Challenge

Find the successor() function for the following sequence
0 xyx
1 xxy
2 xyxy
3 xxyy
4 xyxyyx
5 xxyyxy
6 xyxyxyy
7 xxyxyyyx
8 xyxxyyyxxy
9 xyxyxyyxyxyxy
10 xxyxyyxyxyxyy
11 xyxxyyxyxyxyyyx
12 xyxyxyxyxyxyyyxxy
13 xxyxyxyxyxyyyxyxyxy
14 xyxxyxyxyxyyyxxyxyxyy
15 xyxyxxyxyxyyyxyxyxxyxyyyx
16 xxyyxyxxyxyyyxxyxyyxyxxyyyxxy
17 xyxyxyyxyxxyyyxyxyxxyyxyyxyxyyxyxyxy
18 xxyxyyxyyxyxyyxxyxyyxyxyxyyxyxyyxyxyxyy
19 xyxxyyxyyxyxyyyxxyxxyyxyxyxyyxyxyyxyxyxyyyx

Moses Ilyich Schönfinkel was a Russian logician and mathematician, known for the invention of combinatory logic. After he left Göttingen, Schönfinkel returned to Moscow. By 1927 he was reported to be mentally ill and in a sanatorium.[4][5] His later life was spent in poverty, and he died in Moscow some time in 1942. His papers were burned by his neighbors for heating.[5]