A299416 Solution a( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 3, a(1) = 4; see Comments.

3, 4, 3, 7, 11, 14, 17, 19, 22, 25, 28, 31, 34, 38, 41, 44, 47, 50, 53, 56, 59, 62, 65, 68, 71, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 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] = 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, A299417.

Sequence in context: A141730 A127737 A358125 * A333974 A163108 A350771

Adjacent sequences: A299413 A299414 A299415 * A299417 A299418 A299419

Keyword

nonn,easy

Author

Clark Kimberling, Feb 15 2018

Status

approved