A022426 -id:A022426

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A022424

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

+10
58

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

(list; graph; refs; listen; history; text; internal format)

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.

KEYWORD

nonn

AUTHOR

Clark Kimberling

EXTENSIONS

Edited by Clark Kimberling, Feb 16 2018

STATUS

approved

A299411

Solution b( ) of the complementary equation a(n) = b(n-1) + b(n-2), where a(0) = 2, a(1) = 3; see Comments.

+10
3

1, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56, 58, 59, 61, 62, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 81, 82, 84, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

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] = 2; a[1] = 3; b[0] = 1; b[1] = 4;

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}] (* A022426 *)

Table[b[n], {n, 0, 100}] (* A299411 *)

CROSSREFS

Cf. A022424, A022426.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Feb 14 2018

STATUS

approved

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