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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. 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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 2 16:17 EST 2021. Contains 349445 sequences. (Running on oeis4.)