Number of pairs in generation n of the tree T defined in Comments.

Terms

a(0) =1a(1) =1a(2) =2a(3) =3a(4) =5a(5) =8a(6) =12a(7) =18a(8) =25a(9) =35a(10) =51a(11) =75a(12) =110a(13) =161a(14) =236a(15) =346a(16) =507a(17) =743a(18) =1089a(19) =1596a(20) =2339a(21) =3428a(22) =5024a(23) =7363a(24) =10791a(25) =15815a(26) =23178a(27) =33969a(28) =49784a(29) =72962