A022424 Solution a( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 1, a(1) = 2; see Comments.

1, 2, 7, 9, 11, 14, 18, 22, 25, 28, 31, 33, 36, 39, 41, 44, 47, 50, 53, 56, 59, 62, 66, 69, 72, 75, 78, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 161, 164, 167, 170

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

From the Bode-Harborth-Kimberling link:

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)).

***

In the following guide to solutions a( ) and b( ) of a(n) = b(n-1) + b(n-2), an asterisk (*) indicates that a( ) differs from the indicated A-sequence in one or two initial terms:

(a(n)) (b(n)) a(0) a(1)

A022424 A055563 1 2

A022425 A299407 1 4

A022441* A055562 1 5

A022426 A299411 2 3

A022442* A099467* 2 4

A299416 A299417 3 4

A299418 A299419 3 5

A299420 A299421 4 5

A022441* A055562 1 1

***

Guide to solutions a( ) and b( ) of a(n) = b(n-1) + b(n-2) + b(n-3) for various initial values:

(a(n)) (b(n)) a(0) a(1) a(2)

A299486 A299487 1 2 3

A299488 A299489 1 2 4

A299490 A299491 1 3 5

A299492 A299492 2 4 5

A299494 A299493 2 4 6

A299496 A299494 3 4 5

***

Guide to other complementary equations:

A022427-A022440: a(n) = b(n-1) + b(n-3)

A299531-A299532: a(n) = 2*b(n-1) + b(n-2), a(0) = 1, a(1) = 2

A296220, A299534: a(n) = b(n-1) + 2*b(n-2), a(0) = 1, a(1) = 2

A022437, A299536: a(n) = b(n-1) + b(n-3), a(0) = 1, a(1) = 2, a(2) = 3

A022437, A299538: a(n) = b(n-1) + b(n-3), a(0) = 2, a(1) = 3, a(2) = 4

A022438-A299540: a(n) = b(n-1) + b(n-3), a(0) = 2, a(1) = 3, a(2) = 5

A299541-A299542: a(n) = b(n-1) + b(n-3), a(0) = 2, a(1) = 4, a(2) = 6

A299543-A299544: a(n) = 2*b(n-1) + b(n-2) - b(n-3), a(0) = 1, a(1) = 2, a(2) = 3

A299545-A299546: a(n) = b(n-1) + 2*b(n-2) - b(n-3), a(0) = 1, a(1) = 2, a(2) = 3

A299547: a(n) = b(n-1) + b(n-2) + ... + b(0), a(0) = 1, a(1) = 2, a(2) = 3

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. A055563 (complement), A022425, A299407, A299486-A299494.

Another pair is given in A324142, A324143.

Programs

Mathematica

Fold[Append[#1, Plus @@ Complement[Range[Max@#1 + 3], #1][[{#2, #2 + 1}]]] &, {1, 2}, Range[56]] (* Ivan Neretin, Mar 28 2017 *)

Extensions

Edited by Clark Kimberling, Feb 16 2018