

A298295


Solution a( ) of the complementary equation a(n) = a(0)*b(n) + a(1)*b(n1), 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.


4



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
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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), 113.


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



