A299536 Solution b( ) of the complementary equation a(n) = b(n-1) + b(n-3), where a(0) = 1, a(1) = 2, a(2) = 3; see Comments.

4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 18, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 34, 35, 36, 38, 40, 41, 42, 44, 46, 47, 49, 50, 52, 54, 55, 57, 58, 60, 62, 63, 64, 66, 68, 69, 71, 72, 74, 75, 77, 78, 80, 81, 83, 84, 86, 87, 89, 90, 92, 93, 94, 96, 98, 99, 100

Offset

0,1

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] = 1; a[1] = 2; a[2] = 3; b[0] = 4; 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}]    (* A022427 *)
Table[b[n], {n, 0, 100}]    (* A299536 *)

Crossrefs

Cf. A022424, A022427 (complement).

Sequence in context: A341051 A120181 A016070 * A321025 A047569 A039062

Adjacent sequences: A299533 A299534 A299535 * A299537 A299538 A299539

Keyword

nonn,easy

Author

Clark Kimberling, Feb 24 2018

Status

approved