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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A329835 Beatty sequence for (9+sqrt(101))/10.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120
Offset: 1

Views

Author

Clark Kimberling, Dec 31 2019

Keywords

Comments

Let r = (9+sqrt(101))/10. Then (floor(n*r)) and (floor(n*r + 3r/4)) are a pair of Beatty sequences; i.e., every positive integer is in exactly one of the sequences. See the Guide to related sequences at A329825.

Links

Crossrefs

Cf. A329825, A329836 (complement).

Programs

  • Mathematica
    t = 1/5; r = Simplify[(2 - t + Sqrt[t^2 + 4])/2]; s = Simplify[r/(r - 1)];
Table[Floor[r*n], {n, 1, 200}]   (* A329835 *)
Table[Floor[s*n], {n, 1, 200}]   (* A329836 *)

Formula

a(n) = floor(n*r), where r = (9+sqrt(101))/10.

A329836 Beatty sequence for (11+sqrt(101))/10.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 42, 44, 46, 48, 50, 52, 54, 56, 58, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 82, 84, 86, 88, 90, 92, 94, 96, 98, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 122, 124, 126, 128, 130
Offset: 1

Views

Author

Clark Kimberling, Dec 31 2019

Keywords

Comments

Let r = (9+sqrt(101))/10. Then (floor(n*r)) and (floor(n*r + 3r/4)) are a pair of Beatty sequences; i.e., every positive integer is in exactly one of the sequences. See the Guide to related sequences at A329825.

Links

Crossrefs

Cf. A329825, A329835 (complement).

Programs

  • Mathematica
    t = 1/5; r = Simplify[(2 - t + Sqrt[t^2 + 4])/2]; s = Simplify[r/(r - 1)];
Table[Floor[r*n], {n, 1, 200}]   (* A329835 *)
Table[Floor[s*n], {n, 1, 200}]   (* A329836 *)

Formula

a(n) = floor(n*s), where s = (11+sqrt(101))/10.

A329837 Beatty sequence for (4+sqrt(26))/5.

Original entry on oeis.org

1, 3, 5, 7, 9, 10, 12, 14, 16, 18, 20, 21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 40, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 60, 61, 63, 65, 67, 69, 70, 72, 74, 76, 78, 80, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 100, 101, 103, 105, 107, 109, 111, 112, 114
Offset: 1

Views

Author

Clark Kimberling, Dec 31 2019

Keywords

Comments

Let r = (4+sqrt(26))/5. Then (floor(n*r)) and (floor(n*r + 2r/5)) are a pair of Beatty sequences; i.e., every positive integer is in exactly one of the sequences. See the Guide to related sequences at A329825.

Links

Crossrefs

Cf. A329825, A329838 (complement).

Programs

  • Mathematica
    t = 2/5; r = Simplify[(2 - t + Sqrt[t^2 + 4])/2]; s = Simplify[r/(r - 1)];
Table[Floor[r*n], {n, 1, 200}]   (* A329837 *)
Table[Floor[s*n], {n, 1, 200}]   (* A329838 *)

Formula

a(n) = floor(n*r), where r = (4+sqrt(26))/5.

A329838 Beatty sequence for (6+sqrt(26))/5.

Original entry on oeis.org

2, 4, 6, 8, 11, 13, 15, 17, 19, 22, 24, 26, 28, 31, 33, 35, 37, 39, 42, 44, 46, 48, 51, 53, 55, 57, 59, 62, 64, 66, 68, 71, 73, 75, 77, 79, 82, 84, 86, 88, 91, 93, 95, 97, 99, 102, 104, 106, 108, 110, 113, 115, 117, 119, 122, 124, 126, 128, 130, 133, 135
Offset: 1

Views

Author

Clark Kimberling, Dec 31 2019

Keywords

Comments

Let r = (4+sqrt(26))/5. Then (floor(n*r)) and (floor(n*r + 2r/5)) are a pair of Beatty sequences; i.e., every positive integer is in exactly one of the sequences. See the Guide to related sequences at A329825.

Links

Crossrefs

Cf. A329825, A329837 (complement).

Programs

  • Mathematica
    t = 2/5; r = Simplify[(2 - t + Sqrt[t^2 + 4])/2]; s = Simplify[r/(r - 1)];
Table[Floor[r*n], {n, 1, 200}]   (* A329837 *)
Table[Floor[s*n], {n, 1, 200}]   (* A329838 *)

Formula

a(n) = floor(n*s), where s = (6+sqrt(26))/5.

A329839 Beatty sequence for (-1+sqrt(41))/4.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 21, 22, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 54, 55, 56, 58, 59, 60, 62, 63, 64, 66, 67, 68, 70, 71, 72, 74, 75, 76, 78, 79, 81, 82, 83, 85, 86, 87, 89
Offset: 1

Views

Author

Clark Kimberling, Dec 31 2019

Keywords

Comments

Let r = (-1+sqrt(41))/4. Then (floor(n*r)) and (floor(n*r + 5r/2)) are a pair of Beatty sequences; i.e., every positive integer is in exactly one of the sequences. See the Guide to related sequences at A329825.

Links

Crossrefs

Cf. A329825, A329840 (complement).

Programs

  • Mathematica
    t = 5/2; r = Simplify[(2 - t + Sqrt[t^2 + 4])/2]; s = Simplify[r/(r - 1)];
Table[Floor[r*n], {n, 1, 200}]   (* A329839 *)
Table[Floor[s*n], {n, 1, 200}]   (* A329840 *)

Formula

a(n) = floor(n*r), where r = (-1+sqrt(41))/5.

A329840 Beatty sequence for (9+sqrt(41))/4.

Original entry on oeis.org

3, 7, 11, 15, 19, 23, 26, 30, 34, 38, 42, 46, 50, 53, 57, 61, 65, 69, 73, 77, 80, 84, 88, 92, 96, 100, 103, 107, 111, 115, 119, 123, 127, 130, 134, 138, 142, 146, 150, 154, 157, 161, 165, 169, 173, 177, 180, 184, 188, 192, 196, 200, 204, 207, 211, 215, 219
Offset: 1

Views

Author

Clark Kimberling, Dec 31 2019

Keywords

Comments

Let r = (-1+sqrt(41))/4. Then (floor(n*r)) and (floor(n*r + 5r/2)) are a pair of Beatty sequences; i.e., every positive integer is in exactly one of the sequences. See the Guide to related sequences at A329825.

Links

Crossrefs

Cf. A329825, A329839 (complement).

Programs

  • Mathematica
    t = 5/2; r = Simplify[(2 - t + Sqrt[t^2 + 4])/2]; s = Simplify[r/(r - 1)];
Table[Floor[r*n], {n, 1, 200}]   (* A329839 *)
Table[Floor[s*n], {n, 1, 200}]   (* A329840 *)

Formula

a(n) = floor(n*s), where s = (9+sqrt(41))/4. - corrected by Michael De Vlieger, Aug 27 2021

A329841 Beatty sequence for (7+sqrt(109))/10.

Original entry on oeis.org

1, 3, 5, 6, 8, 10, 12, 13, 15, 17, 19, 20, 22, 24, 26, 27, 29, 31, 33, 34, 36, 38, 40, 41, 43, 45, 47, 48, 50, 52, 54, 55, 57, 59, 61, 62, 64, 66, 68, 69, 71, 73, 74, 76, 78, 80, 81, 83, 85, 87, 88, 90, 92, 94, 95, 97, 99, 101, 102, 104, 106, 108, 109, 111
Offset: 1

Views

Author

Clark Kimberling, Dec 31 2019

Keywords

Comments

Let r = (7+sqrt(109))/10. Then (floor(n*r)) and (floor(n*r + 3r/5)) are a pair of Beatty sequences; i.e., every positive integer is in exactly one of the sequences. See the Guide to related sequences at A329825.

Links

Crossrefs

Cf. A329825, A329842 (complement).

Programs

  • Mathematica
    t = 3/5; r = Simplify[(2 - t + Sqrt[t^2 + 4])/2]; s = Simplify[r/(r - 1)];
Table[Floor[r*n], {n, 1, 200}]   (* A329841 *)
Table[Floor[s*n], {n, 1, 200}]   (* A329842 *)

Formula

a(n) = floor(n*r), where r = (7+sqrt(109))/10.

A329842 Beatty sequence for (13+sqrt(109))/10.

Original entry on oeis.org

2, 4, 7, 9, 11, 14, 16, 18, 21, 23, 25, 28, 30, 32, 35, 37, 39, 42, 44, 46, 49, 51, 53, 56, 58, 60, 63, 65, 67, 70, 72, 75, 77, 79, 82, 84, 86, 89, 91, 93, 96, 98, 100, 103, 105, 107, 110, 112, 114, 117, 119, 121, 124, 126, 128, 131, 133, 135, 138, 140, 142
Offset: 1

Views

Author

Clark Kimberling, Dec 31 2019

Keywords

Comments

Let r = (13+sqrt(109))/10. Then (floor(n*r)) and (floor(n*r + 3r/5)) are a pair of Beatty sequences; i.e., every positive integer is in exactly one of the sequences. See the Guide to related sequences at A329825.

Links

Crossrefs

Cf. A329825, A329841 (complement).

Programs

  • Mathematica
    t = 3/5; r = Simplify[(2 - t + Sqrt[t^2 + 4])/2]; s = Simplify[r/(r - 1)];
Table[Floor[r*n], {n, 1, 200}]   (* A329841 *)
Table[Floor[s*n], {n, 1, 200}]   (* A329842 *)

Formula

a(n) = floor(n*s), where s = (13+sqrt(109))/10.

A329843 Beatty sequence for (1+sqrt(61))/6.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 93, 95, 96
Offset: 1

Views

Author

Clark Kimberling, Jan 02 2020

Keywords

Comments

Let r = (1+sqrt(61))/6. Then (floor(n*r)) and (floor(n*r + 5r/3)) are a pair of Beatty sequences; i.e., every positive integer is in exactly one of the sequences. See the Guide to related sequences at A329825.

Links

Crossrefs

Cf. A329825, A329844 (complement).

Programs

  • Mathematica
    t = 5/3; r = Simplify[(2 - t + Sqrt[t^2 + 4])/2]; s = Simplify[r/(r - 1)];
Table[Floor[r*n], {n, 1, 200}]   (* A329843 *)
Table[Floor[s*n], {n, 1, 200}]   (* A329844 *)

Formula

a(n) = floor(n*r), where r = (1+sqrt(61))/6.

A329844 Beatty sequence for (11+sqrt(61))/6.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 25, 28, 31, 34, 37, 40, 43, 47, 50, 53, 56, 59, 62, 65, 68, 72, 75, 78, 81, 84, 87, 90, 94, 97, 100, 103, 106, 109, 112, 115, 119, 122, 125, 128, 131, 134, 137, 141, 144, 147, 150, 153, 156, 159, 163, 166, 169, 172, 175, 178, 181
Offset: 1

Views

Author

Clark Kimberling, Jan 02 2020

Keywords

Comments

Let r = (1+sqrt(61))/6. Then (floor(n*r)) and (floor(n*r + 5r/3)) are a pair of Beatty sequences; i.e., every positive integer is in exactly one of the sequences. See the Guide to related sequences at A329825.

Links

Crossrefs

Cf. A329825, A329843 (complement).

Programs

  • Mathematica
    t = 5/3; r = Simplify[(2 - t + Sqrt[t^2 + 4])/2]; s = Simplify[r/(r - 1)];
Table[Floor[r*n], {n, 1, 200}]   (* A329843 *)
Table[Floor[s*n], {n, 1, 200}]   (* A329844 *)

Formula

a(n) = floor(n*s), where s = (11+sqrt(61))/6.
Previous Showing 11-20 of 63 results. Next

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