A304813 Solution (a(n)) of the complementary equation a(n) = b(3n) + b(5n); see Comments.

2, 12, 21, 29, 39, 48, 57, 66, 74, 84, 93, 101, 111, 120, 129, 138, 146, 156, 165, 173, 183, 191, 201, 210, 218, 228, 237, 245, 255, 263, 273, 282, 290, 300, 309, 317, 327, 335, 345, 354, 362, 372, 381, 390, 399, 407, 417, 426, 434, 444, 453, 462, 471, 479

Offset

0,1

Comments

Define complementary sequences a(n) and b(n) recursively:

b(n) = least new,

a(n) = b(3n) + b(5n),

where "least new" means the least positive integer not yet placed. Empirically, {a(n) - 8*n: n >= 0} = {2,3} and {7*b(n) - 8*n: n >= 0} = {7,8,9,10,11,12,13,14,15}. See A304799 for a guide to related sequences.

Links

Clark Kimberling, Table of n, a(n) for n = 0..9999

Example

b(0) = 1, so that a(0) = 2.  Since a(1) = b(3) + b(5), we must have a(1) >= 10, so that b(1) = 3, b(2) = 4, b(3) = 5, b(4) = 6, b(5) = 7, and a(1) = 12.

Mathematica

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
h = 3; k = 5; a = {}; b = {1};
AppendTo[a, mex[Flatten[{a, b}], 1]];
Do[Do[AppendTo[b, mex[Flatten[{a, b}], Last[b]]], {k}];
  AppendTo[a, Last[b] + b[[1 + (Length[b] - 1)/k h]]], {500}];
Take[a, 200]  (* A304813 *)
Take[b, 200]  (* A304814 *)
(* Peter J. C. Moses, May 14 2008 *)

Crossrefs

Cf. A304799, A304814.

Sequence in context: A048955 A034049 A271361 * A279424 A073041 A284595

Adjacent sequences: A304810 A304811 A304812 * A304814 A304815 A304816

Keyword

nonn,easy

Author

Clark Kimberling, May 30 2018

Status

approved