Length of n-th iterate of the mapping 00->0010, 10->010, 1-->001, starting with 00.

Terms

a(0) =2a(1) =4a(2) =7a(3) =12a(4) =20a(5) =34a(6) =58a(7) =99a(8) =169a(9) =288a(10) =491a(11) =837a(12) =1428a(13) =2437a(14) =4159a(15) =7099a(16) =12118a(17) =20686a(18) =35313a(19) =60283a(20) =102909a(21) =175676a(22) =299897a(23) =511955a(24) =873961a(25) =1491944a(26) =2546908a(27) =4347844a(28) =7422234a(29) =12670546