A305129 Solution (a(n)) of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n); see Comments.

1, 2, 8, 20, 39, 67, 105, 154, 215, 289, 377, 480, 599, 735, 889, 1062, 1256, 1472, 1711, 1974, 2262, 2576, 2917, 3286, 3684, 4112, 4571, 5062, 5586, 6144, 6737, 7366, 8032, 8736, 9480, 10265, 11092, 11962, 12876, 13835, 14840, 15892, 16992, 18141, 19340

Offset

0,2

Comments

Define sequences a(n) and b(n) recursively, starting with a(0) = 1, a(1) = 2:

b(n) = least new;

a(n) = 2*a(n-1) - a(n-2) + b(n),

where "least new" means the least positive integer not yet placed. It appears that a(n)/a(n-1) -> 1, that {a(n) - a(n-1), n>=1} is unbounded, and that the 3rd difference sequence of (a(n)) consists entirely of 1's and 2's.

Links

Clark Kimberling, Table of n, a(n) for n = 0..10000

Example

b(0) = least not in {a(0), a(1)} = 3;
a(2) = 2*a(1) - a(0) +  b(2) must exceed  = 2*2 -1 + 5 = 8, so that b(0) = 3, b(1) = 4, b(2) = 5, b(3) = 6, b(4) =7, and a(2) = 8.

Mathematica

a = {1, 2}; b = {3, 4, 5};
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
Do[AppendTo[a, 2 Last[a] - a[[-2]] + Last[b]];
  AppendTo[b, mex[Flatten[{a, b}], Last[b]]], {200}]; a
(* Peter J. C. Moses, May 30 2018 *)

Crossrefs

Cf. A305329, A305330.

Sequence in context: A278212 A032767 A294869 * A032633 A294437 A128445

Adjacent sequences: A305126 A305127 A305128 * A305130 A305131 A305132

Keyword

nonn,easy

Author

Clark Kimberling, May 30 2018

Status

approved