A022425 Solution a( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 1, a(1) = 4; see Comments.

1, 4, 5, 9, 13, 15, 18, 21, 23, 26, 30, 33, 36, 39, 42, 46, 49, 52, 55, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 89, 92, 95, 98, 101, 104, 107, 110, 114, 117, 120, 123, 126, 129, 132, 135

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

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

• Ivan Neretin, Table of n, a(n) for n = 0..10000
• J-P. Bode, H. Harborth, C. Kimberling, Complementary Fibonacci sequences, Fibonacci Quarterly 45 (2007), 254-264.

Crossrefs

Cf. A022424, A299407 (complement).

Programs

Mathematica

Fold[Append[#1, Plus @@ Complement[Range[Max@#1 + 3], #1][[{#2, #2 + 1}]]] &, {1, 4}, Range[44]] (* Ivan Neretin, Mar 28 2017 *)

Extensions

Updated by Clark Kimberling, Feb 19 2018