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

0,2

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.

Mathematica

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

Crossrefs

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

Another pair is given in A324142, A324143.

Sequence in context: A003668 A191263 A287359 * A360944 A136498 A297826

Adjacent sequences: A022421 A022422 A022423 * A022425 A022426 A022427

Keyword

nonn

Author

Clark Kimberling

Extensions

Edited by Clark Kimberling, Feb 16 2018

Status

approved