A299418 Solution a( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 3, a(1) = 5; see Comments.

3, 5, 3, 6, 11, 15, 17, 19, 22, 25, 27, 30, 34, 38, 41, 44, 47, 50, 54, 57, 60, 63, 65, 68, 71, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 105, 108, 111, 114, 117, 120, 123, 126, 130, 133, 136, 139, 142, 146, 149, 152, 155, 158, 161, 164, 167, 170, 173

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] = 3; a[1] = 5; 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}]  (* A299418 *)
Table[b[n], {n, 0, 100}]  (* A299419 *)

Crossrefs

Cf. A022424, A299419.

Sequence in context: A324086 A286108 A096438 * A214456 A118271 A260689

Adjacent sequences: A299415 A299416 A299417 * A299419 A299420 A299421

Keyword

nonn,easy

Author

Clark Kimberling, Feb 15 2018

Status

approved