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A298876 Solution (c(n)) of the system of 3 equations in Comments.

%I A298876 #10 Oct 10 2018 10:34:45
%S A298876 3,16,27,43,60,79,100,126,153,182,213,249,289,330,373,418,465,514,565,
%T A298876 624,683,744,807,872,939,1008,1082,1157,1234,1313,1394,1477,1562,1652,
%U A298876 1746,1841,1938,2037,2138,2241,2346,2453,2562,2673,2786,2904,3023,3147
%N A298876 Solution (c(n)) of the system of 3 equations in Comments.
%C A298876 Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
%C A298876 a(n) = least new;
%C A298876 b(n) = a(n) + b(n-1);
%C A298876 c(n) = a(n) + 2 b(n);
%C A298876 where "least new k" means the least positive integer not yet placed.
%C A298876 ***
%C A298876 Do these sequences a,b,c partition the positive integers?  They differ from the corresponding partitioning sequences A298871, A298872, and A298872.  For example, A298872(56) = 2139, whereas A298875(56) = 2138.
%C A298876 Differs from A298873 first at n=56. - _Georg Fischer_, Oct 10 2018
%H A298876 Clark Kimberling, <a href="/A298876/b298876.txt">Table of n, a(n) for n = 0..1000</a>
%e A298876 n:   0    1    2    3    4    5    6    7    8    9
%e A298876 a:   1    4    5    7    8    9   10   12   13   14
%e A298876 b:   2    6   11   18   26   35   45   57   70   84
%e A298876 c:   3   16   27   43   60   30   79  100  126  153
%t A298876 z = 200;
%t A298876 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A298876 a = {1}; b = {2}; c = {3};
%t A298876 Do[{AppendTo[a, mex[Flatten[{a, b, c}], 1]],
%t A298876    AppendTo[b, Last[a] + Last[b]],
%t A298876    AppendTo[c, Last[a] + 2 Last[b]]}, {z}];
%t A298876 Take[a, 100]  (* A298874 *)
%t A298876 Take[b, 100]  (* A298875 *)
%t A298876 Take[c, 100]  (* A298876 *)
%Y A298876 Cf. A299634, A298871, A298874, A298875.
%K A298876 nonn,easy
%O A298876 0,1
%A A298876 _Clark Kimberling_, Apr 19 2018