cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A299540 Solution b( ) of the complementary equation a(n) = b(n-1) + b(n-3), where a(0) = 2, a(1) = 3, a(2) = 5; see Comments.

Original entry on oeis.org

1, 4, 6, 8, 9, 10, 11, 13, 14, 16, 17, 19, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 36, 37, 39, 40, 42, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 72, 73, 74, 76, 78, 79, 80, 82, 84, 85, 87, 88, 90, 92, 93, 94, 96, 98, 99, 100
Offset: 0

Views

Author

Clark Kimberling, Feb 25 2018

Keywords

Comments

From the Bode-Harborth-Kimberling link:
a(n) = b(n-1) + b(n-3) for n > 3;
b(0) = least positive integer not in {a(0),a(1),a(2)};
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

Crossrefs

Cf. A022424, A022438.

Programs

  • Mathematica
    mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 2; a[1] = 3; a[2] = 5; b[0] = 1; b[1] = 4;
a[n_] := a[n] = b[n - 1] + b[n - 3];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}]    (* A022438 *)
Table[b[n], {n, 0, 100}]    (* A299540 *)

A299541 Solution a( ) of the complementary equation a(n) = b(n-1) + b(n-3), where a(0) = 2, a(1) = 4, a(2) = 6; see Comments.

Original entry on oeis.org

2, 4, 6, 6, 10, 13, 16, 19, 21, 25, 27, 31, 33, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 75, 79, 81, 85, 87, 91, 93, 97, 99, 103, 105, 109, 111, 115, 117, 121, 123, 127, 129, 133, 135, 139, 141, 145, 148, 151, 154, 157, 160, 163, 166, 169, 172
Offset: 0

Views

Author

Clark Kimberling, Feb 25 2018

Keywords

Comments

From the Bode-Harborth-Kimberling link:
a(n) = b(n-1) + b(n-3) for n > 3;
b(0) = least positive integer not in {a(0),a(1),a(2)};
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

Crossrefs

Cf. A022424, A299542.

Programs

  • Mathematica
    mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 2; a[1] = 4; a[2] = 6; b[0] = 1; b[1] = 3;
a[n_] := a[n] = b[n - 1] + b[n - 3];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}]    (* A299541 *)
Table[b[n], {n, 0, 100}]    (* A299542 *)

A299542 Solution b( ) of the complementary equation a(n) = b(n-1) + b(n-3), where a(0) = 2, a(1) = 4, a(2) = 6; see Comments.

Original entry on oeis.org

1, 3, 5, 7, 8, 9, 11, 12, 14, 15, 17, 18, 20, 22, 23, 24, 26, 28, 29, 30, 32, 34, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57, 59, 60, 62, 63, 65, 66, 68, 69, 71, 72, 74, 76, 77, 78, 80, 82, 83, 84, 86, 88, 89, 90, 92, 94, 95, 96, 98, 100
Offset: 0

Views

Author

Clark Kimberling, Feb 25 2018

Keywords

Comments

From the Bode-Harborth-Kimberling link:
a(n) = b(n-1) + b(n-3) for n > 3;
b(0) = least positive integer not in {a(0),a(1),a(2)};
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

Crossrefs

Cf. A022424, A299541.

Programs

  • Mathematica
    mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 2; a[1] = 4; a[2] = 6; b[0] = 1; b[1] = 3;
a[n_] := a[n] = b[n - 1] + b[n - 3];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}]    (* A299541 *)
Table[b[n], {n, 0, 100}]    (* A299542 *)

A299543 Solution a( ) of the complementary equation a(n) = 2*b(n-1) + b(n-2) - b(n-3), where a(0) = 1, a(1) = 2, a(2) = 3; see Comments.

Original entry on oeis.org

1, 2, 3, 13, 15, 17, 19, 21, 23, 25, 29, 34, 38, 42, 46, 50, 54, 56, 57, 61, 64, 65, 67, 71, 74, 75, 79, 82, 83, 87, 90, 91, 95, 98, 99, 103, 106, 107, 111, 118, 121, 121, 125, 128, 133, 139, 140, 141, 145, 148, 153, 157, 157, 161, 164, 169, 173, 173, 177
Offset: 0

Views

Author

Clark Kimberling, Feb 25 2018

Keywords

Comments

From the Bode-Harborth-Kimberling link:
a(n) = 2*b(n-1) + b(n-2) - b(n-3) for n > 3;
b(0) = least positive integer not in {a(0),a(1),a(2)};
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

Crossrefs

Cf. A022424, A299544.

Programs

  • Mathematica
    mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; a[2] = 3; b[0] = 4; b[1] = 5;
a[n_] := a[n] = 2 b[n - 1] + b[n - 2] - b[n - 3];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}]    (* A299543 *)
Table[b[n], {n, 0, 100}]    (* A299544 *)

A299544 Solution b( ) of the complementary equation a(n) = 2*b(n-1) + b(n-2) - b(n-3), where a(0) = 1, a(1) = 2, a(2) = 3; see Comments.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 26, 27, 28, 30, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 58, 59, 60, 62, 63, 66, 68, 69, 70, 72, 73, 76, 77, 78, 80, 81, 84, 85, 86, 88, 89, 92, 93, 94, 96, 97, 100, 101
Offset: 0

Views

Author

Clark Kimberling, Feb 25 2018

Keywords

Comments

From the Bode-Harborth-Kimberling link:
a(n) = 2*b(n-1) + b(n-2) - b(n-3) for n > 3;
b(0) = least positive integer not in {a(0),a(1),a(2)};
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

Crossrefs

Cf. A022424, A299543.

Programs

  • Mathematica
    mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; a[2] = 3; b[0] = 4; b[1] = 5;
a[n_] := a[n] = 2 b[n - 1] + b[n - 2] - b[n - 3];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}]    (* A299543 *)
Table[b[n], {n, 0, 100}]    (* A299544 *)

A299545 Solution a( ) of the complementary equation a(n) = b(n-1) + 2*b(n-2) - b(n-3), where a(0) = 1, a(1) = 2, a(2) = 3; see Comments.

Original entry on oeis.org

1, 2, 3, 12, 14, 16, 18, 20, 22, 25, 30, 34, 38, 42, 46, 49, 51, 55, 56, 58, 61, 65, 66, 69, 73, 74, 77, 81, 82, 85, 89, 90, 93, 97, 99, 104, 107, 108, 112, 119, 121, 123, 127, 128, 132, 138, 139, 143, 144, 148, 154, 155, 159, 160, 164, 170, 171, 175, 176
Offset: 0

Views

Author

Clark Kimberling, Mar 01 2018

Keywords

Comments

From the Bode-Harborth-Kimberling link:
a(n) = b(n-1) + 2*b(n-2) - b(n-3) for n > 3;
b(0) = least positive integer not in {a(0),a(1),a(2)};
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

Crossrefs

Cf. A022424, A299546.

Programs

  • Mathematica
    mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; a[2] = 3; b[0] = 4; b[1] = 5;
a[n_] := a[n] = b[n - 1] + 2 b[n - 2] - b[n - 3];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}]    (* A299545 *)
Table[b[n], {n, 0, 100}]    (* A299546 *)

A299546 Solution b( ) of the complementary equation a(n) = b(n-1) + 2*b(n-2) - b(n-3), where a(0) = 1, a(1) = 2, a(2) = 3; see Comments.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 21, 23, 24, 26, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 50, 52, 53, 54, 57, 59, 60, 62, 63, 64, 67, 68, 70, 71, 72, 75, 76, 78, 79, 80, 83, 84, 86, 87, 88, 91, 92, 94, 95, 96, 98, 100, 101
Offset: 0

Views

Author

Clark Kimberling, Mar 01 2018

Keywords

Comments

From the Bode-Harborth-Kimberling link:
a(n) = b(n-1) + 2*b(n-2) - b(n-3) for n > 3;
b(0) = least positive integer not in {a(0),a(1),a(2)};
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

Crossrefs

Cf. A022424, A299545.

Programs

  • Mathematica
    mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; a[2] = 3; b[0] = 4; b[1] = 5;
a[n_] := a[n] = b[n - 1] + 2 b[n - 2] - b[n - 3];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}]    (* A299545 *)
Table[b[n], {n, 0, 100}]    (* A299546 *)

A324142 This sequence and A324143 are a pair of complementary sequences studied by Bode, Harborth, and Kimberling (2007).

Original entry on oeis.org

2, 5, 4, 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, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174, 178
Offset: 1

Views

Author

N. J. A. Sloane, Feb 20 2019

Keywords

Comments

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.

Links

Crossrefs

Cf. A022424, A324143.

Extensions

a(18) and beyond from Lars Blomberg, Feb 21 2019

A022428 a(n) = c(n-1) + c(n-3) where c is the sequence of numbers not in a.

Original entry on oeis.org

1, 2, 4, 9, 12, 14, 17, 19, 23, 26, 29, 33, 36, 39, 42, 45, 47, 51, 53, 57, 59, 62, 65, 67, 71, 73, 77, 79, 83, 85, 89, 92, 95, 98, 101, 104, 107, 110, 113, 116, 119, 123, 125, 129, 132, 135
Offset: 0

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A022424 and references therein.

Programs

  • Mathematica
    Fold[Append[#1, Plus @@ Complement[Range[Max@#1 + 3], #1][[{#2, #2 + 2}]]] &, {1, 2, 4}, Range[43]] (* Ivan Neretin, Mar 29 2017 *)

A022440 a(n) = c(n-1) + c(n-3) where c is the sequence of positive numbers not in a.

Original entry on oeis.org

3, 4, 5, 7, 10, 15, 19, 21, 24, 26, 29, 31, 34, 37, 40, 43, 47, 50, 53, 57, 60, 63, 67, 69, 73, 75, 79, 81, 85, 87, 90, 93, 95, 99, 101, 105, 107, 110, 113, 115, 119, 121, 125, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 169, 172, 175, 179, 181
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

From N. J. A. Sloane, Nov 24 2004: I'm not sure of the minimal hypotheses needed to generate this sequence, but one method that works is the following:
Start with a(1)=3, a(2)=4, a(3)=5, so that we know c(1)=1 and c(2)=2. Let c(3) = x >= 6, so that a(4) = 1+x >= 6 and x=6 is forced, with a(4)=7. Then c(4) >= 8, a(5) >= 10, so definitely c(4)=8 and c(5)=9. From now on the sequence extends easily.

Links

Crossrefs

Cf. A022424 and references therein.

Programs

  • Mathematica
    Fold[Append[#1, Plus @@ Complement[Range[Max@#1 + 3], #1][[{#2, #2 + 2}]]] &, {3, 4, 5}, Range[59]] (* Ivan Neretin, Mar 30 2017 *)

Extensions

More terms from Jon E. Schoenfield, Apr 02 2010
Previous Showing 31-40 of 57 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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