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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 (list; graph; refs; listen; history; text; internal format)
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,

LINKS

Table of n, a(n) for n=0..60.

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

Cf. A304497, A304500, A304502.

Sequence in context: A190429 A239009 A287074 * A283964 A330085 A175884

Adjacent sequences:  A304498 A304499 A304500 * A304502 A304503 A304504

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 19 2018

STATUS

approved

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Last modified May 25 13:00 EDT 2022. Contains 354071 sequences. (Running on oeis4.)