A304501 - OEIS

%I #15 Jun 25 2018 17:30:59

%S 2,4,7,9,12,14,16,18,21,23,25,28,30,32,35,37,39,42,44,46,49,51,54,56,

%T 58,60,63,65,67,70,72,75,77,79,81,84,86,88,91,93,96,98,100,102,105,

%U 107,109,112,114,117,119,121,123,126,128,130,133,135,138,140,142

%N Solution (b(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) = a(n) + 2*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 <= 4,

%C    5 <= 3*b(n) - 7*n <= 8,

%C    4 <=   c(n) - 7*n <= 6,

%e a(0) = 1, b(0) = 2; c(0) = 1 + 2*2 = 5, 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, Last[a] + 2*Last[b]], {z}];

%t Take[a, 100] (* A304500 *)

%t Take[b, 100] (* A304501 *)

%t Take[c, 100] (* A304502 *)

%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. A304497, A304500, A304502.

%K nonn,easy

%O 0,1

%A _Clark Kimberling_, May 19 2018