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 1-4 of 4 results.

A097337 Integer part of the edge of a cube that has space-diagonal n.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 21, 22, 23, 23, 24, 24, 25, 25, 26, 27, 27, 28, 28, 29, 30, 30, 31, 31, 32, 32, 33, 34, 34, 35, 35, 36, 36, 37, 38, 38, 39, 39, 40, 40, 41, 42, 42, 43
Offset: 1

Views

Author

Cino Hilliard, Sep 17 2004

Keywords

Comments

The first few terms are the same as A038128. However, A038128 is generated by Euler's constant = 0.5772156649015328606065120901..., which is close but not equal to 1/sqrt(3) = 0.5773502691896257645091487805..., which generates this sequence. Euler/(1/sqrt(3)) = 0.9997668585341064519813571911... and the equality fails in the 97th term.
The integers k such that a(k) = a(k+1) give A054406. - Michel Marcus, Nov 01 2021

References

  • The Universal Encyclopedia of Mathematics, English translation, 1964, p. 155.

Links

Crossrefs

Cf. A020760 (1/sqrt(3)), A054406.

Programs

  • PARI
    f(n) = for(x=1,n,s=x\sqrt(3);print1(s","));s
  • PARI
    a(n)=sqrtint(n^2\3) \\ Charles R Greathouse IV, Nov 01 2021

Formula

Let L be the length of the edges. Then sqrt(2)*L is the diagonal of a face. Whence n^2 = 2*L^2 + L^2, or n = sqrt(3)*L and L = n/sqrt(3).

A059555 Beatty sequence for 1 + gamma A001620.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 18, 20, 22, 23, 25, 26, 28, 29, 31, 33, 34, 36, 37, 39, 41, 42, 44, 45, 47, 48, 50, 52, 53, 55, 56, 58, 59, 61, 63, 64, 66, 67, 69, 70, 72, 74, 75, 77, 78, 80, 82, 83, 85, 86, 88, 89, 91, 93, 94, 96, 97, 99, 100, 102, 104, 105, 107, 108
Offset: 1

Views

Author

Mitch Harris, Jan 22 2001

Keywords

Comments

Let r = gamma (the Euler constant, 0.5772...). When {k*r, k >= 1} is jointly ranked with the positive integers, A059555(n) is the position of n and A059556(n) is the position of n*r. - Clark Kimberling, Oct 21 2014

Links

Crossrefs

Beatty complement is A059556.

Programs

  • Magma
    R:=RealField(100); [Floor((1+EulerGamma(R))*n): n in [1..100]]; // G. C. Greubel, Aug 27 2018
  • Maple
    A001620 := proc(n)
        floor((1+gamma)*n) ;
end proc:
seq(A001620(n),n=1..50) ; # R. J. Mathar, Nov 11 2011
  • Mathematica
    t = N[Table[k*EulerGamma, {k, 1, 200}]]; u = Union[Range[200], t]
Flatten[Table[Flatten[Position[u, n]], {n, 1, 100}]]  (* A059556 *)
Flatten[Table[Flatten[Position[u, t[[n]]]], {n, 1, 100}]] (* A059555 *)
(* Clark Kimberling, Oct 21 2014 *)
  • PARI
    { default(realprecision, 100); b=1 + Euler; for (n = 1, 2000, write("b059555.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

Formula

a(n) = n + A038128(n).

A184977 a(n) = Sum_{k=1..n} floor(k*gamma) where gamma is Euler's constant (A001620).

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 13, 17, 22, 27, 33, 39, 46, 54, 62, 71, 80, 90, 100, 111, 123, 135, 148, 161, 175, 190, 205, 221, 237, 254, 271, 289, 308, 327, 347, 367, 388, 409, 431, 454, 477, 501, 525, 550, 575, 601, 628, 655, 683, 711, 740, 770, 800, 831, 862, 894, 926, 959, 993, 1027, 1062
Offset: 1

Views

Author

Michel Lagneau, Mar 27 2011

Keywords

Comments

a(n) = A183143(n) for n = 1..96 where A183143(n) is the sequence floor(1/r) + floor(2/r) + ... + floor(n/r) and r = sqrt(3). It is interesting to note that a(n)/n^2 converges to gamma/2.
gamma = 0.57721566490153286060651209... (A002852)
1/sqrt(3) = 0.577350269189625764509148... (A020760)
Starts to differ from A183143 at a(97). - R. J. Mathar, Aug 28 2025

Links

Crossrefs

Cf. A001620, A038128.

Programs

  • Magma
    R:=RealField(100); [(&+[Floor(k*EulerGamma(R)): k in [1..n]]): n in [1..50]]; // G. C. Greubel, Aug 27 2018
  • Maple
    with(numtheory):Digits:=500:s:=0:c:=evalf(gamma(0)):for n from 1 to 100 do:
  s:=s+floor(n*c):printf(`%d, `,s):od:
  • Mathematica
    Table[Sum[Floor[k*EulerGamma], {k, 1, n}], {n, 50}] (* G. C. Greubel, Jun 02 2017 *)
  • PARI
    a(n) = sum(k=1, n, floor(k*Euler)); \\ Michel Marcus, Apr 02 2017

Formula

Partial sums of A038128.

Extensions

Name edited by Jon E. Schoenfield, Apr 02 2017

A263574 Beatty sequence for 1/sqrt(3) - log(phi)/3575 where phi is the golden ratio, A001622.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 4, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 21, 22, 23, 23, 24, 24, 25, 25, 26, 27, 27, 28, 28, 29, 30, 30, 31, 31, 32, 32, 33, 34, 34, 35, 35, 36, 36, 37, 38, 38, 39, 39, 40, 40, 41, 42, 42, 43
Offset: 0

Views

Author

Karl V. Keller, Jr., Oct 21 2015

Keywords

Comments

The number 1/sqrt(3) - log(phi)/3575 (=0.577215664483...) is an approximation to Euler's constant (A001620) (=0.577215664901...).
M. Hudson found a similar Euler-Mascheroni constant approximation (see link), 1/sqrt(3)-1/7429 (=0.57721566157...).

Examples

			For n=9, floor(9*(0.577215664483)) = floor(5.194940980347) = 5.

Links

Crossrefs

Cf. A001620, A020760 (1/sqrt(3)), A038128 (Beatty sequence for Euler's constant), A097337 (Beatty sequence for 1/sqrt(3)).

Programs

  • Magma
    phi:= (1+Sqrt(5))/2; [Floor(n*(1/Sqrt(3) - Log(phi)/3575)): n in [0..100]]; // G. C. Greubel, Sep 05 2018
  • Mathematica
    Table[Floor[n (1/Sqrt@ 3 - Log[GoldenRatio]/3575)], {n, 0, 75}] (* Michael De Vlieger, Nov 12 2015 *)
  • PARI
    {phi = (1+sqrt(5))/2}; vector(100, n, n--; floor(n*(1/sqrt(3) - log(phi)/3575))) \\ G. C. Greubel, Sep 05 2018
  • Python
    from sympy import floor, log, sqrt
for n in range(0,101):print(floor(n*(1/sqrt(3)-log(1/2+sqrt(5)/2)/3575)),end=', ')

Formula

a(n) = floor(n*(1/sqrt(3) - log(phi)/3575)).
a(n) = A038128(n) for n < 58628.
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.