A298295 Solution a( ) of the complementary equation a(n) = a(0)*b(n) + a(1)*b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments.

1, 2, 13, 16, 19, 22, 25, 28, 31, 34, 38, 43, 47, 52, 56, 61, 65, 70, 74, 79, 83, 88, 92, 97, 101, 106, 109, 113, 118, 121, 124, 128, 133, 136, 140, 145, 148, 151, 155, 160, 163, 167, 172, 175, 178, 182, 187, 190, 194, 199, 202, 205, 209, 214, 217, 221, 226

Offset

0,2

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values.

Links

Clark Kimberling, Table of n, a(n) for n = 0..1000

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Example

a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, so that a(2) = 13.
Complement: (3,4,5,6,7,8,9,10,11,12,14,15,17,...)

Mathematica

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[0]*b[n] + a[1]*b[n - 1]
Table[{a[n],
   b[n + 1] = mex[Flatten[Map[{a[#], b[#]} &, Range[0, n]]], b[n - 0]]}, {n, 2, 1010}];
Table[a[n], {n, 0, 150}]  (* A298295 *)
Table[b[n], {n, 0, 150}]  (* A298296 *)
(* Peter J. C. Moses, Jan 16 2018 *)

Crossrefs

Cf. A298296, A297830, A298000.

Sequence in context: A041645 A318999 A032453 * A257636 A258317 A318911

Adjacent sequences: A298292 A298293 A298294 * A298296 A298297 A298298

Keyword

nonn,easy

Author

Clark Kimberling, Feb 09 2018

Status

approved