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A304499
Solution (c(n)) of the system of complementary equations defined in Comments.
3
4, 11, 19, 25, 32, 40, 46, 52, 61, 67, 74, 82, 88, 95, 103, 109, 115, 124, 130, 137, 145, 151, 160, 166, 172, 178, 187, 193, 200, 208, 214, 221, 229, 235, 241, 250, 256, 263, 271, 277, 284, 292, 298, 304, 313, 319, 326, 334, 340, 349, 355, 361, 367, 376, 382
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) = 2*a(n) + b(n),
where "least new k" means the least positive integer not yet placed. The three sequences partition the positive integers. Empirically:
1 <= 3*a(n) - 7*n <= 5,
5 <= 3*b(n) - 7*n <= 8,
3 <= c(n) - 7*n <= 6.
EXAMPLE
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.
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, 2 Last[a] + Last[b]], {z}];
Take[a, 100] (* A304497 *)
Take[b, 100] (* A304498 *)
Take[c, 100] (* A304499 *)
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 16 2018
STATUS
approved