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This pair of sequences is the first example in the Bode, Harborth, and Kimberling (2007) paper. See A022424 for a list of a large number of other pairs of complementary sequences based on the same paper.

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)).

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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

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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

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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