Number of integers in n-th generation of tree T(1/3) defined in Comments.

Terms

a(0) =1a(1) =1a(2) =1a(3) =1a(4) =2a(5) =2a(6) =2a(7) =4a(8) =4a(9) =5a(10) =8a(11) =9a(12) =12a(13) =16a(14) =20a(15) =26a(16) =34a(17) =44a(18) =57a(19) =74a(20) =97a(21) =125a(22) =162a(23) =212a(24) =272a(25) =356a(26) =462a(27) =597a(28) =780a(29) =1010