Length of n-th iterate of the mapping 00->0010, 01->010, 10->000, starting with 00.

Terms

a(0) =2a(1) =4a(2) =7a(3) =12a(4) =22a(5) =39a(6) =69a(7) =122a(8) =217a(9) =386a(10) =687a(11) =1224a(12) =2181a(13) =3883a(14) =6910a(15) =12305a(16) =21906a(17) =38991a(18) =69421a(19) =123609a(20) =220110a(21) =391929a(22) =697931a(23) =1242873a(24) =2213237a(25) =3941323a(26) =7018445a(27) =12497917a(28) =22255832a(29) =39632383