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%I #16 Jul 05 2018 06:22:03
%S 1,3,6,8,10,13,15,17,20,22,24,27,29,31,34,36,38,41,43,45,48,50,53,55,
%T 57,59,62,64,66,69,71,73,76,78,80,83,85,87,90,92,94,97,99,101,104,106,
%U 108,111,113,116,118,120,122,125,127,129,132,134,136,139,141
%N Solution (a(n)) of the system of complementary equations defined in Comments.
%C Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1:
%C a(n) = least new,
%C b(n) = least new,
%C c(n) = 2*a(n) + b(n),
%C where "least new k" means the least positive integer not yet placed. The three sequences partition the positive integers. Empirically, for all n >= 0,
%C 1 <= 3*a(n) - 7*n <= 5,
%C 5 <= 3*b(n) - 7*n <= 8,
%C 3 <= c(n) - 7*n <= 6.
%e a(0) = 1, b(0) = 2; c(0) = 2*1 + 2 = 4, so that a(1) = 3, so that b(1) = 4, so that c(1) = 11.
%t z = 300;
%t mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
%t a = {}; b = {}; c = {};
%t Do[AppendTo[a,
%t mex[Flatten[{a, b, c}], If[Length[a] == 0, 1, Last[a]]]];
%t AppendTo[b, mex[Flatten[{a, b, c}], Last[a]]];
%t AppendTo[c, 2 Last[a] + Last[b]], {z}];
%t Take[a, 100] (* A304497 *)
%t Take[b, 100] (* A304498 *)
%t Take[c, 100] (* A304499 *)
%t Grid[{Join[{"n"}, Range[0, 20]], Join[{"a(n)"}, Take[a, 21]],
%t Join[{"b(n)"}, Take[b, 21]], Join[{"c(n)"}, Take[c, 21]]},
%t Alignment -> ".", Dividers -> {{2 -> Red, -1 -> Blue}, {2 -> Red, -1 -> Blue}}]
%t (* _Peter J. C. Moses_, Apr 26 2018 *)
%Y Cf. A304498, A304499, A304500.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, May 16 2018
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