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A330500 a(n) = a(n-1) + a(floor(n/3)), a(1) = a(2) = 1.

%I A330500 #22 Dec 16 2019 20:05:20
%S A330500 1,1,2,3,4,5,6,7,9,11,13,16,19,22,26,30,34,39,44,49,55,61,67,74,81,88,
%T A330500 97,106,115,126,137,148,161,174,187,203,219,235,254,273,292,314,336,
%U A330500 358,384,410,436,466,496,526,560,594,628,667,706,745,789,833,877
%N A330500 a(n) = a(n-1) + a(floor(n/3)), a(1) = a(2) = 1.
%C A330500 Also, the number of finite sequences b(1..r) satisfying b(1) = 1 and b(i+1) >= 3*b(i) and b(r) <= n.
%e A330500 For n = 10 the 11 sequences enumerated are (1), (1,3), (1,4), (1,5), (1,6), (1,7), (1,8), (1,9), (1,10), (1,3,9), (1,3,10).
%p A330500 a:= proc(n) option remember;
%p A330500       `if`(n<2, n, a(n-1)+a(iquo(n, 3)))
%p A330500     end:
%p A330500 seq(a(n), n=1..75);  # _Alois P. Heinz_, Dec 16 2019
%t A330500 Nest[Append[#1, #1[[-1]] + #1[[Floor[#2/3] ]] ] & @@ {#, Length@ # + 1} &, {1, 1}, 57] (* _Michael De Vlieger_, Dec 16 2019 *)
%Y A330500 An analog of A033485.
%K A330500 nonn
%O A330500 1,3
%A A330500 _Jeffrey Shallit_, Dec 16 2019