

A298296


Solution b( ) 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



3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87
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OFFSET

0,1


COMMENTS

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values; b(n)b(n1) is in {1,2} for all n >= 1.


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,...) = (b(n)).


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. A298295, A297830, A298000.
Sequence in context: A304816 A298005 A216442 * A099474 A105048 A026511
Adjacent sequences: A298293 A298294 A298295 * A298297 A298298 A298299


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Feb 09 2018


STATUS

approved



