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A296502 Solution (b(n)) of the system of 3 complementary equations in Comments.

%I A296502 #4 Apr 24 2018 12:10:52
%S A296502 2,5,14,32,53,77,104,134,170,209,254,302,353,407,464,524,587,653,722,
%T A296502 794,869,950,1034,1121,1211,1304,1403,1505,1610,1718,1829,1943,2060,
%U A296502 2180,2303,2429,2558,2690,2825,2966,3110,3257,3407,3560,3716,3878,4043,4211
%N A296502 Solution (b(n)) of the system of 3 complementary equations in Comments.
%C A296502 Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 4:
%C A296502 a(n) = least new;
%C A296502 b(n) = a(n-1)+c(n-1);
%C A296502 c(n) = 2 a(n) + b(n);
%C A296502 where "least new k" means the least positive integer not yet placed. The sequences a,b,c partition the positive integers.
%H A296502 Clark Kimberling, <a href="/A296502/b296502.txt">Table of n, a(n) for n = 0..1000</a>
%e A296502 n:   0   1    2    3    4    5    6    7   8     9
%e A296502 a:   1   3    6    7    8    9   10   12  13    15
%e A296502 b:   2   5   14   32   53   77  104  134  170  209
%e A296502 c:   4  11   26   46   69   95  124  158  196  239
%t A296502 z = 300;
%t A296502 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A296502 a = {1}; b = {2}; c = {4}; n = 1;
%t A296502 Do[{n++, AppendTo[a, mex[Flatten[{a, b, c}], 1]],
%t A296502    AppendTo[b, a[[n - 1]] + c[[n - 1]]],
%t A296502    AppendTo[c, 2 Last[a] + Last[b]]}, {z}];
%t A296502 Take[a, 100]  (* A296484 *)
%t A296502 Take[b, 100]  (* A296502 *)
%t A296502 Take[c, 100]  (* A297149 *)
%Y A296502 Cf. A299634, A296484, A297149.
%K A296502 nonn,easy
%O A296502 0,1
%A A296502 _Clark Kimberling_, Apr 24 2018