A059539 -id:A059539 - cp's OEIS frontend

A081129 Differences of Beatty sequence for cube root of 3.

1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2

Offset: 0

Benoit Cloitre, Apr 16 2003

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A059539, A081117, A081147, A081168.

A081129:= func< n | Floor((n+1)*3^(1/3)) - Floor(n*3^(1/3)) >;
[A081129(n): n in [0..120]]; // G. C. Greubel, Jan 15 2024

Differences[Floor[Range[0,110]Surd[3,3]]] (* Harvey P. Dale, Apr 06 2022 *)

a(n)=floor((n+1)*3^(1/3))-floor(n*3^(1/3))

def A081129(n): return floor((n+1)*3^(1/3)) - floor(n*3^(1/3))
[A081129(n) for n in range(121)] # G. C. Greubel, Jan 15 2024

a(n) = floor((n+1)*3^(1/3)) - floor(n*3^(1/3)).

A038129 Beatty sequence for cube root of 2.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 32, 34, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 78, 79, 80, 81, 83, 84, 85, 86, 88, 89

Offset: 0

Felice Russo

nonn

Index entries for sequences related to Beatty sequences

Cf. A059539.

Floor[Range[0,90]Surd[2,3]] (* Harvey P. Dale, Mar 24 2019 *)

PARI
```
a(n)=floor(n*2^(1/3))
```

from sympy import integer_nthroot
def A038129(n): return integer_nthroot(2*n**3,3)[0] # Chai Wah Wu, Mar 17 2021

a(n)=floor(n*1.2599210498..)

a(n)=floor(n*2^(1/3)) - Benoit Cloitre, Apr 16 2003

More terms from Benoit Cloitre, Apr 16 2003

A059540 Beatty sequence for 3^(1/3)/(3^(1/3)-1).

Original entry on oeis.org

3, 6, 9, 13, 16, 19, 22, 26, 29, 32, 35, 39, 42, 45, 48, 52, 55, 58, 61, 65, 68, 71, 75, 78, 81, 84, 88, 91, 94, 97, 101, 104, 107, 110, 114, 117, 120, 123, 127, 130, 133, 136, 140, 143, 146, 150, 153, 156, 159, 163, 166, 169, 172, 176, 179, 182, 185, 189, 192

Offset: 1

Mitch Harris, Jan 22 2001

Harry J. Smith, Table of n, a(n) for n = 1..2000
Aviezri S. Fraenkel, Jonathan Levitt, 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 A059539.

Cf. A072365.

Floor[Range[100]/(1 - 3^(-1/3))] (* Paolo Xausa, Jul 17 2024 *)

{ default(realprecision, 100); b=3^(1/3)/(3^(1/3) - 1); for (n = 1, 2000, write("b059540.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = floor(n/(1 - A072365)). - Paolo Xausa, Jul 17 2024

A248188 Numbers k such that A248186(k+1) = A248186(k) + 1.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 04 2014

a(n) = A059539(n+2) = [3^(1/3)*(n+2)] for n = 1..655, but a(656) = 948 = A059539(658)-1.

			The difference sequence of A248186 is (0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, ...), so that A248187 = (1, 2, 3, 6, 9, 13, 16, 19, 22,...) and A248188 = (4, 5, 7, 8, 10, 11, 12, 14, 15, 17,...), the complement of A248186.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A248186, A248187, A248183.

$MaxExtraPrecision = Infinity;
z = 800; p[k_] := p[k] = Sum[1/(h*(h + 1)*(h + 2)*(h + 3)), {h, 1, k}];
N[Table[1/18 - p[n], {n, 1, z/10}]]
f[n_] := f[n] = Select[Range[z], 1/18 - p[#] < 1/n^3 &, 1]
u = Flatten[Table[f[n], {n, 1, z}]]   (* A248186 *)
Flatten[Position[Differences[u], 0]]  (* A248187 *)
Flatten[Position[Differences[u], 1]]  (* A248188 *)

A187574 Rank transform of the sequence floor(n*3^(1/3)); complement of A187575.

Original entry on oeis.org

1, 3, 5, 6, 9, 10, 12, 14, 15, 17, 19, 21, 23, 25, 26, 28, 30, 31, 34, 35, 37, 39, 41, 42, 45, 46, 48, 50, 51, 54, 55, 57, 59, 61, 62, 64, 66, 67, 70, 71, 73, 75, 77, 79, 80, 82, 84, 86, 87, 90, 91, 93, 95, 96, 98, 100, 102, 104, 106, 107, 109, 111, 112, 115, 116, 118, 120, 122, 124, 125, 127, 129, 131, 132, 135, 136, 138, 140, 142, 143, 145, 147, 149, 151, 152, 154, 156, 158, 160

Offset: 1

Clark Kimberling, Mar 11 2011

nonn

See A187224.

Cf. A187224, A187575, A059539.

seqA = Table[Floor[n*3^(1/3)], {n, 1, 220}] (*A059539*)
seqB = Table[n, {n, 1, 220}];(*A000027*)
jointRank[{seqA_,
   seqB_}] := {Flatten@Position[#1, {_, 1}],
    Flatten@Position[#1, {_, 2}]} &[
  Sort@Flatten[{{#1, 1} & /@ seqA, {#1, 2} & /@ seqB}, 1]];
limseqU =
FixedPoint[jointRank[{seqA, #1[[1]]}] &,
   jointRank[{seqA, seqB}]][[1]] (*A187574*)
Complement[Range[Length[seqA]], limseqU]  (*A187575*)
(* Peter J. C. Moses, Mar 11 2011 *)

A307513 Beatty sequence for 1/log(2).

Original entry on oeis.org

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, 75, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 90, 92, 93, 95, 96, 98, 99, 100, 102, 103, 105, 106, 108, 109, 111, 112, 113, 115

Offset: 1

R. J. Mathar, Apr 12 2019

Very similar to A059539 because A002581 is close to A007525.

Cf. A007525.

a(n) = floor(n*A007525).

A166986(n) = 2*a(n+2)-4.

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)

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A081129 Differences of Beatty sequence for cube root of 3.

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A038129 Beatty sequence for cube root of 2.

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A059540 Beatty sequence for 3^(1/3)/(3^(1/3)-1).

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A248188 Numbers k such that A248186(k+1) = A248186(k) + 1.

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A187574 Rank transform of the sequence floor(n*3^(1/3)); complement of A187575.

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A307513 Beatty sequence for 1/log(2).

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

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