a(n) is the number of subsequences {s(k)} of {1,2,3,...n} such that s(k+1)-s(k) is 1 or 3.

Terms

a(0) =1a(1) =3a(2) =6a(3) =11a(4) =19a(5) =31a(6) =49a(7) =76a(8) =116a(9) =175a(10) =262a(11) =390a(12) =578a(13) =854a(14) =1259a(15) =1853a(16) =2724a(17) =4001a(18) =5873a(19) =8617a(20) =12639a(21) =18534a(22) =27174a(23) =39837a(24) =58396a(25) =85596a(26) =125460a(27) =183884a(28) =269509a(29) =394999