A059540 -id:A059540 - cp's OEIS frontend

A059539 Beatty sequence for 3^(1/3).

1, 2, 4, 5, 7, 8, 10, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 25, 27, 28, 30, 31, 33, 34, 36, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 51, 53, 54, 56, 57, 59, 60, 62, 63, 64, 66, 67, 69, 70, 72, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 87, 89, 90, 92, 93, 95, 96, 98, 99, 100

Offset: 1

Mitch Harris, Jan 22 2001

Harry J. Smith, Table of n, a(n) for n = 1..2000
Aviezri S. Fraenkel, Jonathan Levitt, and Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Beatty complement is A059540.
Partial sums of A081129.
Cf. A002581.

Floor[Range[100]*CubeRoot[3]] (* Paolo Xausa, Jul 05 2024 *)

{ default(realprecision, 100); b=3^(1/3); for (n = 1, 2000, write("b059539.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 27 2009

from sympy import integer_nthroot
def A059539(n): return integer_nthroot(3*n**3,3)[0] # Chai Wah Wu, Mar 16 2021

a(n) = floor(n*A002581). - R. J. Mathar, Apr 12 2019

A249179 First row of spectral array W(3^(1/3)).

Original entry on oeis.org

1, 3, 4, 9, 12, 29, 41, 94, 135, 306, 441, 997, 1437, 3251, 4688, 10602, 15290, 34574, 49864, 112751, 162615, 367699, 530313, 1199127, 1729440, 3910553, 5639993, 12752965

Offset: 1

Colin Barker, Dec 03 2014

3^(1/3) = 1.442249570307408382321638310780109588391869253499350577546416...

The sequence is generated from the Beatty sequence (A059539) and from the complement of the Beatty sequence (A059540) for 3^(1/3).

A. Fraenkel and C. Kimberling, Generalized Wythoff arrays, shuffles and interspersions, Discrete Mathematics 126 (1994) 137-149.

Cf. A059539, A059540.

\\ Row i of the generalized Wythoff array W(h),
\\   where h is an irrational number between 1 and 2,
\\   and m is the number of terms in the vectors b and c.
row(h, i, m) = {
  if(h<=1 || h>=2, print("Invalid value for h"); return);
  my(
    b=vector(m, n, floor(n*h)),       \\ Beatty sequence for h
    c=vector(m, n, floor(n*h/(h-1))), \\ Complement of b
    w=[b[b[i]], c[b[i]]],
    j=3
  );
  while(1,
    if(j%2==1,
      if(w[j-1]<=#b, w=concat(w, b[w[j-1]]), return(w))
    ,
      if(w[j-2]<=#c, w=concat(w, c[w[j-2]]), return(w))
    );
    j++
  )
}
allocatemem(10^9)
default(realprecision, 100)
row(3^(1/3), 1, 10^7)

cp's OEIS Frontend

A059539 Beatty sequence for 3^(1/3).

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