A299417 Solution b( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 3, a(1) = 4; see Comments.

1, 2, 5, 6, 8, 9, 10, 12, 13, 15, 16, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 37, 39, 40, 42, 43, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 74, 75, 77, 78, 80, 81, 83, 84, 86, 87, 89, 90, 92, 93, 95, 96, 98, 99

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] = 3; a[1] = 4; b[0] = 1; b[1] = 2;
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}]  (* A299416 *)
Table[b[n], {n, 0, 100}]  (* A299417 *)

Crossrefs

Cf. A022424, A299416.

Sequence in context: A224778 A286727 A090946 * A047442 A005781 A157850

Adjacent sequences: A299414 A299415 A299416 * A299418 A299419 A299420

Keyword

nonn,easy,changed

Author

Clark Kimberling, Feb 15 2018

Status

approved