A299420 Solution a( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 4, a(1) = 5; see Comments.

4, 5, 3, 8, 13, 16, 19, 21, 23, 26, 29, 32, 35, 38, 42, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 81, 84, 87, 89, 92, 95, 98, 101, 104, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 146, 149, 152, 155, 158, 162, 165, 168, 171, 174

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] = 4; 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}]  (* A299420 *)
Table[b[n], {n, 0, 100}]  (* A299421 *)

Crossrefs

Cf. A022424, A299421.

Sequence in context: A328238 A170929 A245085 * A019836 A353314 A020503

Adjacent sequences: A299417 A299418 A299419 * A299421 A299422 A299423

Keyword

nonn,easy

Author

Clark Kimberling, Feb 16 2018

Status

approved