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A304501
Solution (b(n)) of the system of complementary equations defined in Comments.
3
2, 4, 7, 9, 12, 14, 16, 18, 21, 23, 25, 28, 30, 32, 35, 37, 39, 42, 44, 46, 49, 51, 54, 56, 58, 60, 63, 65, 67, 70, 72, 75, 77, 79, 81, 84, 86, 88, 91, 93, 96, 98, 100, 102, 105, 107, 109, 112, 114, 117, 119, 121, 123, 126, 128, 130, 133, 135, 138, 140, 142
OFFSET
0,1
COMMENTS
Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1:
a(n) = least new,
b(n) = least new,
c(n) = a(n) + 2*b(n),
where "least new k" means the least positive integer not yet placed. The three sequences partition the positive integers. Empirically, for all n >= 0:
1 <= 3*a(n) - 7*n <= 4,
5 <= 3*b(n) - 7*n <= 8,
4 <= c(n) - 7*n <= 6,
EXAMPLE
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.
MATHEMATICA
z = 300;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {}; b = {}; c = {};
Do[AppendTo[a,
mex[Flatten[{a, b, c}], If[Length[a] == 0, 1, Last[a]]]];
AppendTo[b, mex[Flatten[{a, b, c}], Last[a]]];
AppendTo[c, Last[a] + 2*Last[b]], {z}];
Take[a, 100] (* A304500 *)
Take[b, 100] (* A304501 *)
Take[c, 100] (* A304502 *)
Grid[{Join[{"n"}, Range[0, 20]], Join[{"a(n)"}, Take[a, 21]],
Join[{"b(n)"}, Take[b, 21]], Join[{"c(n)"}, Take[c, 21]]},
Alignment -> ".", Dividers -> {{2 -> Red, -1 -> Blue}, {2 -> Red, -1 -> Blue}}]
(* Peter J. C. Moses, Apr 26 2018 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, May 19 2018
STATUS
approved