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

Author

Clark Kimberling, Feb 09 2018

Keywords

• nonn
• easy

Comments

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

Examples

			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,...)
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..1000
• Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Crossrefs

Cf. A298296, A297830, A298000.

Programs

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 *)