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A298343 a(n) = a(n-1) + a(n-2) + a([2n/3]), where a(0) = 1, a(1) = 2, a(2) = 3.

%I A298343 #6 Apr 26 2022 06:45:06
%S A298343 1,2,3,8,14,30,58,102,190,350,598,1050,1838,3078,5266,8942,14806,
%T A298343 24798,41442,68078,112598,185942,303806,498690,817302,1330798,2172898,
%U A298343 3545138,5759478,9372694,15244770,24730062,40160774,65194642,105659222,171352554,277829078
%N A298343 a(n) = a(n-1) + a(n-2) + a([2n/3]), where a(0) = 1, a(1) = 2, a(2) = 3.
%C A298343 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 A298343 Clark Kimberling, <a href="/A298343/b298343.txt">Table of n, a(n) for n = 0..1000</a>
%p A298343 A298343 := proc(n)
%p A298343     option remember ;
%p A298343     if n <=2 then
%p A298343         n+1 ;
%p A298343     else
%p A298343         procname(n-1)+procname(n-2)+procname(floor(2*n/3)) ;
%p A298343     end if;
%p A298343 end proc:
%p A298343 seq(A298343(n),n=0..80) ; # _R. J. Mathar_, Apr 26 2022
%t A298343 a[0] = 1; a[1] = 2; a[2] = 3;
%t A298343 a[n_] := a[n] = a[n - 1] + a[n - 2] + a[Floor[2n/3]];
%t A298343 Table[a[n], {n, 0, 30}]  (* A298343 *)
%Y A298343 Cf. A001622, A000045, A298338.
%K A298343 nonn,easy
%O A298343 0,2
%A A298343 _Clark Kimberling_, Feb 09 2018