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A298349 a(n) = a(n-1) + a(n-2) + a([(n+1)/2]), where a(0) = 1, a(1) = 2, a(2) = 3.

%I A298349 #4 Feb 10 2018 09:49:49
%S A298349 1,2,3,8,14,30,52,96,162,288,480,820,1352,2268,3716,6146,10024,16458,
%T A298349 26770,43708,70958,115486,187264,304102,492718,799088,1294074,2096878,
%U A298349 3394668,5497692,8898506,14406222,23314752,37737432,61068642,98832844,159928256
%N A298349 a(n) = a(n-1) + a(n-2) + a([(n+1)/2]), where a(0) = 1, a(1) = 2, a(2) = 3.
%C A298349 a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio (A001622), so that (a(n)) has the growth rate of the Fibonacci numbers (A000045). See A298338 for a guide to related sequences.
%H A298349 Clark Kimberling, <a href="/A298349/b298349.txt">Table of n, a(n) for n = 0..1000</a>
%t A298349 a[0] = 1; a[1] = 2; a[2] = 3;
%t A298349 a[n_] := a[n] = a[n - 1] + a[n - 2] + a[Floor[(n+1)/2]];
%t A298349 Table[a[n], {n, 0, 30}]  (* A298349 *)
%Y A298349 Cf. A001622, A000045, A298338.
%K A298349 nonn,easy
%O A298349 0,2
%A A298349 _Clark Kimberling_, Feb 10 2018