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.

A054406 Beatty sequence for (3+sqrt 3)/2; complement of A022838.

Original entry on oeis.org

2, 4, 7, 9, 11, 14, 16, 18, 21, 23, 26, 28, 30, 33, 35, 37, 40, 42, 44, 47, 49, 52, 54, 56, 59, 61, 63, 66, 68, 70, 73, 75, 78, 80, 82, 85, 87, 89, 92, 94, 97, 99, 101, 104, 106, 108, 111, 113, 115, 118, 120, 123, 125, 127, 130, 132, 134, 137, 139, 141, 144, 146
Offset: 1

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Author

Eric W. Weisstein

Keywords

Comments

Numbers k such that A194979(k+1) = A194979(k). - Clark Kimberling, Dec 02 2014

Links

Crossrefs

Cf. A194143 (partial sums), A182778 (even bisection), A184799 (prime terms).
Cf. A022838 (complement), A026255.
Cf. A194979.

Programs

A003512 A Beatty sequence: floor(n*(sqrt(3) + 2)).

Original entry on oeis.org

3, 7, 11, 14, 18, 22, 26, 29, 33, 37, 41, 44, 48, 52, 55, 59, 63, 67, 70, 74, 78, 82, 85, 89, 93, 97, 100, 104, 108, 111, 115, 119, 123, 126, 130, 134, 138, 141, 145, 149, 153, 156, 160, 164, 167, 171, 175, 179, 182, 186
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A003511 (complement), A019973 (sqrt(3)+2).

Programs

  • Maple
    Digits := 60: A003512 := proc(n) trunc( evalf( n*(sqrt(3)+2) )); end;
  • Mathematica
    Table[Floor[n (Sqrt@ 3 + 2)], {n, 50}] (* Michael De Vlieger, Oct 08 2016 *)
  • Python
    from gmpy2 import isqrt
def A003512(n):
    return 2*n + int(isqrt(3*n**2))  # Chai Wah Wu, Oct 08 2016

Formula

a(n) = floor(n*(sqrt(3)+2)). - Michel Marcus, Jan 05 2015
For n >= 0, a(n) = 2n + largest integer m such that m^2 <= 3*n^2. - Chai Wah Wu, Oct 08 2016
From Miko Labalan, Dec 03 2016: (Start)
For n > 0, a(n) = 4*floor(n*(sqrt(3)-1)) + 3*floor(n*(2-sqrt(3))) + 3;
a(0) = 0, a(n) = a(n - 1) + A182778(n) - A182778(n - 1) - 1.
(End)

A182777 Beatty sequence for 3-sqrt(3).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 38, 39, 40, 41, 43, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73, 74, 76, 77, 78, 79, 81, 82, 83, 84, 86
Offset: 1

Views

Author

Clark Kimberling, Nov 30 2010

Keywords

Comments

(1) 3 is the only number x for which the numbers r=x-sqrt(x) and s=x+sqrt(x) satisfy the Beatty equation
1/r + 1/s = 1.
(2) Let u=2-sqrt(3) and v=1. Jointly rank {j*u} and {k*v} as in the first comment at A182760; a(n) is the position of n*u.
(3) The complement of A182777 is A182778, which gives the positions of the natural numbers k in the joint ranking.

Links

Crossrefs

Cf. A182760, A182778.

Programs

  • Magma
    [Floor(n*(3-Sqrt(3))): n in [1..80]]; // Vincenzo Librandi, Oct 25 2011
  • Mathematica
    Table[Floor[(3-Sqrt[3]) n], {n, 68}]
  • PARI
    vector(80, n, floor(n*(3-sqrt(3)))) \\ G. C. Greubel, Nov 23 2018
  • Sage
    [floor(n*(3-sqrt(3))) for n in (1..80)] # G. C. Greubel, Nov 23 2018

Formula

a(n) = floor(n*(3-sqrt(3))).

Extensions

Typo in formula by Vincenzo Librandi, Oct 25 2011
Showing 1-3 of 3 results.

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

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