A299538 Solution b( ) of the complementary equation a(n) = b(n-1) + b(n-3), where a(0) = 2, a(1) = 3, a(2) = 4; see Comments.

1, 5, 6, 8, 9, 10, 11, 12, 14, 16, 17, 19, 21, 23, 24, 26, 27, 29, 30, 32, 33, 34, 36, 37, 39, 40, 41, 43, 44, 46, 47, 48, 50, 52, 53, 54, 56, 58, 59, 60, 62, 64, 65, 67, 68, 70, 72, 73, 74, 76, 78, 79, 81, 82, 84, 86, 87, 88, 90, 92, 93, 95, 96, 98, 99, 101

Offset

0,2

Comments

From the Bode-Harborth-Kimberling link:

a(n) = b(n-1) + b(n-3) for n > 3;

b(0) = least positive integer not in {a(0),a(1),a(2)};

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

Table of n, a(n) for n=0..65.

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; a[2] = 4; b[0] = 1; b[1] = 5;
a[n_] := a[n] = b[n - 1] + b[n - 3];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}]    (* A022437 *)
Table[b[n], {n, 0, 100}]    (* A299538 *)

Crossrefs

Cf. A022424, A299537.

Sequence in context: A205705 A343616 A270653 * A201512 A242290 A296562

Adjacent sequences: A299535 A299536 A299537 * A299539 A299540 A299541

Keyword

nonn,easy

Author

Clark Kimberling, Feb 25 2018

Status

approved