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A298871 Solution (a(n)) of the system of 3 complementary equations in Comments.

%I A298871 #4 Apr 18 2018 19:49:25
%S A298871 1,4,5,7,8,9,10,12,13,14,15,17,19,20,21,22,23,24,25,28,29,30,31,32,33,
%T A298871 34,36,37,38,39,40,41,42,44,46,47,48,49,50,51,52,53,54,55,56,58,59,61,
%U A298871 62,63,64,65,66,67,68,69,71,72,73,74,75,76,77,78,80,81
%N A298871 Solution (a(n)) of the system of 3 complementary equations in Comments.
%C A298871 Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:
%C A298871 a(n) = least new;
%C A298871 b(n) = least new k >= a(n) + b(n-1);
%C A298871 c(n) = a(n) + 2 b(n);
%C A298871 where "least new k" means the least positive integer not yet placed. The sequences a,b,c partition the positive integers.
%H A298871 Clark Kimberling, <a href="/A298871/b298871.txt">Table of n, a(n) for n = 0..1000</a>
%e A298871 n:   0    1    2    3    4    5    6    7    8   9
%e A298871 a:   1    4    5    7    8    9   10   12   13   14
%e A298871 b:   2    6   11   18   26   35   45   57   70   84
%e A298871 c:   3   16   27   43   60   30   79  100  126  153
%t A298871 z = 400;
%t A298871 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A298871 a = {1}; b = {2}; c = {3};
%t A298871 Do[{AppendTo[a, mex[Flatten[{a, b, c}], 1]],
%t A298871    AppendTo[b, mex[Flatten[{a, b, c}], Last[a] + Last[b]]],
%t A298871    AppendTo[c, Last[a] + 2 Last[b]]}, {z}];
%t A298871 Take[a, 100] (* A298871 *)
%t A298871 Take[b, 100] (* A298872 *)
%t A298871 Take[c, 100] (* A298873 *)
%Y A298871 Cf. A299634, A298872, A298873, A298874.
%K A298871 nonn,easy
%O A298871 0,2
%A A298871 _Clark Kimberling_, Apr 18 2018