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A297291 Solution (a(n)) of the system of 3 complementary equations in Comments.

%I A297291 #4 Apr 25 2018 08:32:51
%S A297291 1,4,5,9,12,13,16,17,21,22,27,28,31,32,37,38,41,44,47,48,51,52,57,58,
%T A297291 61,62,67,68,71,72,77,78,81,84,85,89,90,93,97,98,101,104,107,108,111,
%U A297291 112,117,118,121,122,127,128,131,132,137,138,141,144,147,148
%N A297291 Solution (a(n)) of the system of 3 complementary equations in Comments.
%C A297291 Define sequences a(n), b(n), c(n) recursively:
%C A297291 a(n) = least new;
%C A297291 b(n) = least new > = a(n) + 2;
%C A297291 c(n) = a(n) + b(n) - 2;
%C A297291 where "least new k" means the least positive integer not yet placed.
%C A297291 ***
%C A297291 The sequences a,b,c partition the positive integers.
%C A297291 ***
%C A297291 Conjectures:  for n >=0,
%C A297291 0 <= 5*n + 4 - 2*a(n) <= 5,
%C A297291 0 <= 5*n + 8 - 2*b(n) <= 4,
%C A297291 0 <= c(n) - 5n <= 4.
%H A297291 Clark Kimberling, <a href="/A297291/b297291.txt">Table of n, a(n) for n = 0..1000</a>
%e A297291 n:   0   1   2   3   4   5   6   7   8   9  10
%e A297291 a:   1   4   5   9  12  13  16  17  21  27  28
%e A297291 b:   3   6   7  11  14  15  19  20  23  25  29
%e A297291 c:   2   8  10  18  24  26  33  35  42  45  54
%t A297291 z = 300;
%t A297291 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A297291 a = b = c = {};
%t A297291 Do[{AppendTo[a,
%t A297291     mex[Flatten[{a, b, c}], If[Length[a] == 0, 1, Last[a]]]],
%t A297291    AppendTo[b, mex[Flatten[{a, b, c}], Last[a] + 2]],
%t A297291    AppendTo[c, Last[a] + Last[b] - 2]}, {z}];
%t A297291 Take[a, 100]  (* A297291 *)
%t A297291 Take[b, 100]  (* A297292 *)
%t A297291 Take[c, 100]  (* A297293 *)
%t A297291 (* _Peter J. C. Moses_,  Apr 23 2018 *)
%Y A297291 Cf. A299634, A297292, A297293.
%K A297291 nonn,easy
%O A297291 0,2
%A A297291 _Clark Kimberling_, Apr 24 2018