A305329 Solution (a(n)) of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-2); see Comments.

1, 2, 6, 14, 27, 47, 75, 112, 159, 217, 287, 370, 468, 582, 713, 862, 1030, 1218, 1427, 1658, 1912, 2190, 2493, 2822, 3179, 3565, 3981, 4428, 4907, 5419, 5965, 6546, 7163, 7817, 8509, 9240, 10011, 10823, 11677, 12574, 13515, 14501, 15533, 16613, 17742, 18921

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-2),

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(0) must exceed  = 2*2 -1 + 3 = 6, so that b(0) = 3, b(1) = 4, b(2) = 5, and a(2) = 6.

Mathematica

a = {1, 2}; b = {3, 4, 5};
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
Do[AppendTo[a, 2 Last[a] - a[[-2]] + b[[-3]]];
  AppendTo[b, mex[Flatten[{a, b}], Last[b]]], {120}]; a
 (* Peter J. C. Moses, May 30 2018 *)

Crossrefs

Cf. A305129, A305330.

Sequence in context: A101586 A178080 A121968 * A161212 A256058 A294867

Adjacent sequences: A305326 A305327 A305328 * A305330 A305331 A305332

Keyword

nonn,easy

Author

Clark Kimberling, Jun 02 2018

Status

approved