A299411 Solution b( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 2, a(1) = 3; see Comments.

1, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56, 58, 59, 61, 62, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 81, 82, 84, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100

Offset

0,2

Comments

a(n) = b(n-1) + b(n-2) for n > 2;

b(0) = least positive integer not in {a(0),a(1)};

b(n) = least positive integer not in {a(0),...,a(n),b(0),...b(n-1)} for n > 1.

Note that (b(n)) is strictly increasing and is the complement of (a(n)).

See A022424 for a guide to related sequences.

Links

Clark Kimberling, Table of n, a(n) for n = 0..2000

J-P. Bode, H. Harborth, C. Kimberling, Complementary Fibonacci sequences, Fibonacci Quarterly 45 (2007), 254-264.

Mathematica

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 2; a[1] = 3; b[0] = 1; b[1] = 4;
a[n_] := a[n] = b[n - 1] + b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}]  (* A022426 *)
Table[b[n], {n, 0, 100}]  (* A299411 *)

Crossrefs

Cf. A022424, A022426.

Sequence in context: A167524 A356721 A114308 * A079000 A357122 A047509

Adjacent sequences: A299408 A299409 A299410 * A299412 A299413 A299414

Keyword

nonn,easy

Author

Clark Kimberling, Feb 14 2018

Status

approved