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A297149 Solution (c(n)) of the system of 3 complementary equations in Comments.

%I A297149 #4 Apr 24 2018 12:10:59
%S A297149 4,11,26,46,69,95,124,158,196,239,286,336,389,445,504,566,631,699,770,
%T A297149 844,923,1006,1092,1181,1273,1370,1471,1575,1682,1792,1905,2021,2140,
%U A297149 2262,2387,2515,2646,2780,2919,3062,3208,3357,3509,3664,3824,3988,4155,4325
%N A297149 Solution (c(n)) of the system of 3 complementary equations in Comments.
%C A297149 Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 4:
%C A297149 a(n) = least new;
%C A297149 b(n) = a(n-1)+c(n-1);
%C A297149 c(n) = 2 a(n) + b(n);
%C A297149 where "least new k" means the least positive integer not yet placed. The sequences a,b,c partition the positive integers.
%H A297149 Clark Kimberling, <a href="/A297149/b297149.txt">Table of n, a(n) for n = 0..1000</a>
%e A297149 n:   0   1    2    3    4    5    6    7   8     9
%e A297149 a:   1   3    6    7    8    9   10   12  13    15
%e A297149 b:   2   5   14   32   53   77  104  134  170  209
%e A297149 c:   4  11   26   46   69   95  124  158  196  239
%t A297149 z = 300;
%t A297149 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A297149 a = {1}; b = {2}; c = {4}; n = 1;
%t A297149 Do[{n++, AppendTo[a, mex[Flatten[{a, b, c}], 1]],
%t A297149    AppendTo[b, a[[n - 1]] + c[[n - 1]]],
%t A297149    AppendTo[c, 2 Last[a] + Last[b]]}, {z}];
%t A297149 Take[a, 100]  (* A296484 *)
%t A297149 Take[b, 100]  (* A296502 *)
%t A297149 Take[c, 100]  (* A297149 *)
%Y A297149 Cf. A299634, A296484, A296502.
%K A297149 nonn,easy
%O A297149 0,1
%A A297149 _Clark Kimberling_, Apr 24 2018