A304500 Solution (a(n)) of the system of complementary equations defined in Comments.

1, 3, 6, 8, 10, 13, 15, 17, 19, 22, 24, 27, 29, 31, 33, 36, 38, 40, 43, 45, 48, 50, 52, 55, 57, 59, 62, 64, 66, 69, 71, 73, 76, 78, 80, 82, 85, 87, 90, 92, 94, 97, 99, 101, 104, 106, 108, 111, 113, 115, 118, 120, 122, 125, 127, 129, 132, 134, 136, 139, 141

Offset

0,2

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

Robert Israel, Table of n, a(n) for n = 0..10000

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.

Maple

S:= {$1..200}:
for n from 0 do
  if nops(S) < 2 then break fi;
  a[n]:= min(S);
  S:= S minus {a[n]};
  b[n]:= min(S);
  c[n]:= a[n]+2*b[n];
  S:= S minus {b[n], c[n]}
od:
seq(A[i]i=0..n-1); # Robert Israel, Jul 30 2018

Mathematica

z = 301;
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, A304501, A304502.

Sequence in context: A323028 A189471 A169863 * A047282 A304497 A189937

Adjacent sequences: A304497 A304498 A304499 * A304501 A304502 A304503

Keyword

nonn,easy

Author

Clark Kimberling, May 19 2018

Status

approved