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A297293 Solution (c(n)) of the system of 3 complementary equations in Comments.

%I A297293 #10 May 01 2018 03:00:54
%S A297293 2,8,10,18,24,26,33,35,42,45,54,56,63,66,74,76,82,88,94,96,102,105,
%T A297293 114,116,123,125,134,136,142,145,154,156,162,168,170,178,180,186,194,
%U A297293 196,202,208,214,216,222,225,234,236,243,246,254,256,262,265,274,276
%N A297293 Solution (c(n)) of the system of 3 complementary equations in Comments.
%C A297293 Define sequences a(n), b(n), c(n) recursively:
%C A297293 a(n) = least new;
%C A297293 b(n) = least new > = a(n) + 2;
%C A297293 c(n) = a(n) + b(n) - 2;
%C A297293 where "least new k" means the least positive integer not yet placed.
%C A297293 ***
%C A297293 The sequences a,b,c partition the positive integers.
%C A297293 ***
%C A297293 Conjectures:  for n >= 0,
%C A297293 0 <= 5*n + 4 - 2*a(n) <= 5,
%C A297293 0 <= 5*n + 8 - 2*b(n) <= 4,
%C A297293 0 <= c(n) - 5n <= 4.
%H A297293 Clark Kimberling, <a href="/A297293/b297293.txt">Table of n, a(n) for n = 0..1000</a>
%e A297293 n:   0   1   2   3   4   5   6   7   8   9  10
%e A297293 a:   1   4   5   9  12  13  16  17  21  27  28
%e A297293 b:   3   6   7  11  14  15  19  20  23  25  29
%e A297293 c:   2   8  10  18  24  26  33  35  42  45  54
%t A297293 z = 300;
%t A297293 mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t A297293 a = b = c = {};
%t A297293 Do[{AppendTo[a,
%t A297293     mex[Flatten[{a, b, c}], If[Length[a] == 0, 1, Last[a]]]],
%t A297293    AppendTo[b, mex[Flatten[{a, b, c}], Last[a] + 2]],
%t A297293    AppendTo[c, Last[a] + Last[b] - 2]}, {z}];
%t A297293 Take[a, 100]  (* A297291 *)
%t A297293 Take[b, 100]  (* A297292 *)
%t A297293 Take[c, 100]  (* A297293 *)
%t A297293 (* _Peter J. C. Moses_, Apr 23 2018 *)
%Y A297293 Cf. A299634, A297291, A297292.
%K A297293 nonn,easy
%O A297293 0,1
%A A297293 _Clark Kimberling_, Apr 24 2018