A299407 Solution b( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 1, a(1) = 4; see Comments.

2, 3, 6, 7, 8, 10, 11, 12, 14, 16, 17, 19, 20, 22, 24, 25, 27, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 44, 45, 47, 48, 50, 51, 53, 54, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 90, 91, 93, 94, 96, 97, 99, 100

Offset

0,1

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] = 1; a[1] = 4; b[0] = 2; b[1] = 3;
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}]  (* A022425 *)
Table[b[n], {n, 0, 100}]  (* A299407 *)

Crossrefs

Cf. A022424, A022425.

Sequence in context: A028733 A028789 A038100 * A028757 A184798 A332021

Adjacent sequences: A299404 A299405 A299406 * A299408 A299409 A299410

Keyword

nonn,easy

Author

Clark Kimberling, Feb 14 2018

Status

approved