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A297292 Solution (b(n)) of the system of 3 complementary equations in Comments.

%I A297292 #12 May 01 2018 03:00:48
%S A297292 3,6,7,11,14,15,19,20,23,25,29,30,34,36,39,40,43,46,49,50,53,55,59,60,
%T A297292 64,65,69,70,73,75,79,80,83,86,87,91,92,95,99,100,103,106,109,110,113,
%U A297292 115,119,120,124,126,129,130,133,135,139,140,143,146,149,150
%N A297292 Solution (b(n)) of the system of 3 complementary equations in Comments.
%C A297292 Define sequences a(n), b(n), c(n) recursively:
%C A297292 a(n) = least new;
%C A297292 b(n) = least new > = a(n) + 2;
%C A297292 c(n) = a(n) + b(n) - 2;
%C A297292 where "least new k" means the least positive integer not yet placed.
%C A297292 ***
%C A297292 The sequences a,b,c partition the positive integers.
%C A297292 ***
%C A297292 Conjectures:  for n >= 0,
%C A297292 0 <= 5*n + 4 - 2*a(n) <= 5,
%C A297292 0 <= 5*n + 8 - 2*b(n) <= 4,
%C A297292 0 <= c(n) - 5n <= 4.
%H A297292 Clark Kimberling, <a href="/A297292/b297292.txt">Table of n, a(n) for n = 0..1000</a>
%e A297292 n:   0   1   2   3   4   5   6   7   8   9  10
%e A297292 a:   1   4   5   9  12  13  16  17  21  27  28
%e A297292 b:   3   6   7  11  14  15  19  20  23  25  29
%e A297292 c:   2   8  10  18  24  26  33  35  42  45  54
%t A297292 z = 300;
%t A297292 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A297292 a = b = c = {};
%t A297292 Do[{AppendTo[a,
%t A297292     mex[Flatten[{a, b, c}], If[Length[a] == 0, 1, Last[a]]]],
%t A297292    AppendTo[b, mex[Flatten[{a, b, c}], Last[a] + 2]],
%t A297292    AppendTo[c, Last[a] + Last[b] - 2]}, {z}];
%t A297292 Take[a, 100]  (* A297291 *)
%t A297292 Take[b, 100]  (* A297292 *)
%t A297292 Take[c, 100]  (* A297293 *)
%t A297292 (* _Peter J. C. Moses_, Apr 23 2018 *)
%Y A297292 Cf. A299634, A297291, A297293.
%K A297292 nonn,easy
%O A297292 0,1
%A A297292 _Clark Kimberling_, Apr 24 2018