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.

Showing 1-3 of 3 results.

A257984 Nonhomogeneous Beatty sequence: ceiling((n - 1/2)*Pi).

Original entry on oeis.org

2, 5, 8, 11, 15, 18, 21, 24, 27, 30, 33, 37, 40, 43, 46, 49, 52, 55, 59, 62, 65, 68, 71, 74, 77, 81, 84, 87, 90, 93, 96, 99, 103, 106, 109, 112, 115, 118, 121, 125, 128, 131, 134, 137, 140, 143, 147, 150, 153, 156, 159, 162, 165, 169, 172, 175, 178, 181, 184
Offset: 1

Views

Author

Clark Kimberling, Jun 15 2015

Keywords

Comments

Let r = Pi, s = r/(r-1), and t = 1/2. Let R be the ordered set {floor[(n + t)*r] : n is an integer} and let S be the ordered set {floor[(n - t)*s : n is an integer}; thus,
R = (..., -10, -9, -7, -6, -4, -3, -1, 0, 2, 3, 5, 6, 8, ...);
S = (..., -15, -11, -8, -5, -2, 1, 4, 7, 10, 14, 17, 20, ...).
By Fraenkel's theorem (Theorem XI in the cited paper); R and S partition the integers.
R is the set of integers n such that (cos n)*(cos(n + 1)) < 0;
S is the set of integers n such that (cos n)*(cos(n + 1)) > 0.
A246046 = (2,3,5,6,8,...), positive terms of R;
A062389 = (1,4,7,10,14,17,...), positive terms of S;
A258048 = (1,3,4,6,7,9,10,...), - (negative terms of R);
A257984 = (2,5,8,11,15,...), - (negative terms of S).
A062389 and A246046 partition the positive integers, and A258048 and A257984 partition the positive integers.

Links

Crossrefs

Cf. A258048 (complement), A246046, A062380, A258833.

Programs

  • Mathematica
    Table[Ceiling[(n - 1/2) Pi], {n, 1, 120}] (* A257984 *)
Table[Ceiling[(n + 1/2) Pi/(Pi - 1)], {n, 0, 120}]  (* A258048 *)

Formula

a(n) = ceiling((n - 1/2)*Pi).

A258048 Nonhomogeneous Beatty sequence: a(n) = ceiling((n + 1/2)*Pi/(Pi - 1)).

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 12, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 50, 51, 53, 54, 56, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97
Offset: 0

Views

Author

Clark Kimberling, Jun 15 2015

Keywords

Comments

See A257984.

Links

Crossrefs

Cf. A257984 (complement), A246046, A062380, A258833.

Programs

  • Mathematica
    Table[Ceiling[(n - 1/2) Pi], {n, 1, 120}] (* A257984 *)
Table[Ceiling[(n + 1/2) Pi/(Pi - 1)], {n, 0, 120}]  (* A258048 *)

Formula

a(n) = ceiling((n + 1/2)*Pi/(Pi - 1)).

A258834 Nonhomogeneous Beatty sequence: a(n) = ceiling((n - 1/4)*(2 + sqrt(2))).

Original entry on oeis.org

0, 3, 6, 10, 13, 17, 20, 24, 27, 30, 34, 37, 41, 44, 47, 51, 54, 58, 61, 65, 68, 71, 75, 78, 82, 85, 88, 92, 95, 99, 102, 105, 109, 112, 116, 119, 123, 126, 129, 133, 136, 140, 143, 146, 150, 153, 157, 160, 164, 167, 170, 174, 177, 181, 184, 187, 191, 194
Offset: 0

Views

Author

Clark Kimberling, Jun 12 2015

Keywords

Comments

Complement of A258833.
See A258833 for more comments.

Links

Crossrefs

Cf. A258833 (complement), A184580, A184581.

Programs

  • Magma
    [Ceiling((n-1/4)*(2+Sqrt(2))): n in [0..80]]; // Vincenzo Librandi, Jun 13 2015
  • Mathematica
    r = Sqrt[2]; s = r/(r - 1);
Table[Ceiling[(n + 1/4) r], {n, 0, 100}] (* A258833 *)
Table[Ceiling[(n - 1/4) s], {n, 0, 100}] (* A258834 *)
  • PARI
    vector(60, n, ceil((n-1/4)*(2+sqrt(2)))) \\ G. C. Greubel, Aug 19 2018

Formula

a(n) = ceiling((n - 1/4)*(2 + sqrt(2))) = floor((n - 1/4)*(2 + sqrt(2)) + 1).

Extensions

Corrected by Michel Dekking, Sep 19 2019
Showing 1-3 of 3 results.

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