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A296484 Solution (a(n)) of the system of 3 complementary equations in Comments.

%I A296484 #4 Apr 24 2018 12:10:46
%S A296484 1,3,6,7,8,9,10,12,13,15,16,17,18,19,20,21,22,23,24,25,27,28,29,30,31,
%T A296484 33,34,35,36,37,38,39,40,41,42,43,44,45,47,48,49,50,51,52,54,55,56,57,
%U A296484 58,59,60,61,62,63,64,65,66,67,68,70,71,72,73,74,75,76
%N A296484 Solution (a(n)) of the system of 3 complementary equations in Comments.
%C A296484 Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 4:
%C A296484 a(n) = least new;
%C A296484 b(n) = a(n-1)+c(n-1);
%C A296484 c(n) = 2 a(n) + b(n);
%C A296484 where "least new k" means the least positive integer not yet placed. The sequences a,b,c partition the positive integers.
%H A296484 Clark Kimberling, <a href="/A296484/b296484.txt">Table of n, a(n) for n = 0..1000</a>
%e A296484 n:   0   1    2    3    4    5    6    7   8     9
%e A296484 a:   1   3    6    7    8    9   10   12  13    15
%e A296484 b:   2   5   14   32   53   77  104  134  170  209
%e A296484 c:   4  11   26   46   69   95  124  158  196  239
%t A296484 z = 300;
%t A296484 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A296484 a = {1}; b = {2}; c = {4}; n = 1;
%t A296484 Do[{n++, AppendTo[a, mex[Flatten[{a, b, c}], 1]],
%t A296484    AppendTo[b, a[[n - 1]] + c[[n - 1]]],
%t A296484    AppendTo[c, 2 Last[a] + Last[b]]}, {z}];
%t A296484 Take[a, 100]  (* A296484 *)
%t A296484 Take[b, 100]  (* A296502 *)
%t A296484 Take[c, 100]  (* A297149 *)
%Y A296484 Cf. A299634, A296502, A297149.
%K A296484 nonn,easy
%O A296484 0,2
%A A296484 _Clark Kimberling_, Apr 24 2018