A316711 -id:A316711 - cp's OEIS frontend

A276383 Complement of A158919: complementary Beatty sequence to the Beatty sequence defined by the tribonacci constant tau = A058265.

2, 4, 6, 8, 10, 13, 15, 17, 19, 21, 24, 26, 28, 30, 32, 35, 37, 39, 41, 43, 46, 48, 50, 52, 54, 56, 59, 61, 63, 65, 67, 70, 72, 74, 76, 78, 81, 83, 85, 87, 89, 92, 94, 96, 98, 100, 102, 105, 107, 109, 111, 113, 116, 118, 120, 122, 124, 127, 129, 131, 133, 135, 138, 140, 142, 144, 146, 149, 151, 153, 155, 157, 159, 162, 164, 166, 168, 170, 173, 175, 177, 179, 181, 184, 186, 188, 190, 192, 195, 197, 199, 201, 203, 205, 208, 210, 212, 214, 216, 219, 221, 223, 225, 227, 230, 232, 234, 236, 238, 241, 243

Offset: 1

N. J. A. Sloane, Sep 02 2016

nonn

This is the Beatty sequence for tau_prime = 2.191487883953118747061354268227517294...,
defined by 1/tau + 1/tau_prime = 1.
Differs from A172278 at n = 162, 209, 256, 303, 324, ...
Note that Beatty sequences do not normally include 0 - see the classic pair A000201, A001950. - N. J. A. Sloane, Oct 19 2018
Note that the tribonacci numbers T = A000073 related to the ternary sequence A080843 lead to the three complementary sequences for the nonnegative integers AT(n) = A278040(n), BT(n) = A278039(n) and CT(n) = A278041(n). - Wolfdieter Lang, Sep 08 2018

			Comments from _Wolfdieter Lang_, Sep 08 2018 (Start):
The complementary sequences A158919 and A276383 begin:
n:       1 2 3 4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 ...
A158919: 1 3 5 7  9 11 12 14 16 18 20 22 23 25 27 29 31 33 34 36 ...
A276383: 2 4 6 8 10 13 15 17 19 21 24 26 28 30 32 35 37 39 41 43 ...
--------------------------------------------------------------------
The complementary sequences AT, BT and CT begin:
n:  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15  16  17  18  19 ...
AT: 1  5  8 12 14 18 21 25 29 32 36 38 42 45 49 52  56  58  62  65 ...
BT: 0  2  4  6  7  9 11 13 15 17 19 20 22 24 26 28  30  31  33  35 ...
CT: 3 10 16 23 27 34 40 47 54 60 67 71 78 84 91 97 104 108 115 121 ...
(End)

R. J. Mathar, Table of n, a(n) for n = 1..1000
Wikipedia, Beatty Sequence

Cf. A000201, A001950, A058265, A158919.
Cf. also A000073, A080843, A276383, A278039, A278040, A278041, A316711.
Similar to but strictly different from A172278.

A276383 := proc(n)
    Tau := (1/3)*(1+(19+3*sqrt(33))^(1/3)+(19-3*sqrt(33))^(1/3));
    taupr := 1/(1-1/Tau) ;
    floor(n*taupr) ;
end proc: # R. J. Mathar, Sep 04 2016
a:=proc(n) local s,t; t:=evalf(solve(x^3-x^2-x-1=0,x),120)[1]; s:=t/(t-1); floor(n*s) end; seq(a(n),n=0..70); # Muniru A Asiru, Oct 16 2018

a(n) = floor(n*tau_prime), with tau_prime = tau/(tau - 1), where tau is the tribonacci constant A058265.

tau_prime = (1 + (19 + 3*sqrt(33))^(1/3) + (19 - 3*sqrt(33))^(1/3)) / (-2 + (19 + 3*sqrt(33))^(1/3) + (19 - 3*sqrt(33))^(1/3)). - Wolfdieter Lang, Sep 08 2018

Edited by N. J. A. Sloane, Oct 19 2018 at the suggestion of Georg Fischer

A317202 Decimal expansion of 3 + (t^2+t^4)/2, where t = 0.543689... (A192918) is the real root of x^3 + x^2 + x = 1.

Original entry on oeis.org

3, 1, 9, 1, 4, 8, 7, 8, 8, 3, 9, 5, 3, 1, 1, 8, 7, 4, 7, 0, 6, 1, 3, 5, 4, 2, 6, 8, 2, 2, 7, 5, 1, 7, 2, 9, 3, 4, 7, 4, 6, 9, 1, 0, 2, 1, 8, 7, 4, 2, 8, 8, 0, 9, 1, 0, 0, 9, 7, 8, 1, 3, 3, 8, 6, 1, 7, 6, 8, 5, 9, 4, 8, 0, 0, 4, 9, 7, 0, 1, 4, 6, 1, 1, 1

Offset: 1

N. J. A. Sloane, Aug 05 2018

The minimal polynomial of this constant is 2*x^3 - 12*x^2 + 22*x - 13, and it is its unique real root. - Amiram Eldar, May 30 2023

			3.191487883953118747061354268227517293474691...

Bo Tan and Zhi-Ying Wen, Some properties of the Tribonacci sequence, European Journal of Combinatorics, 28 (2007) 1703-1719. See Theorem 4.5.

Cf. A316711, A192918.

RealDigits[x /. FindRoot[2*x^3 - 12*x^2 + 22*x - 13, {x, 3}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, May 30 2023 *)

A357463 Decimal expansion of the real root of 2x^3 + 2x - 1.

Original entry on oeis.org

4, 2, 3, 8, 5, 3, 7, 9, 9, 0, 6, 9, 7, 8, 3, 2, 7, 1, 3, 7, 8, 0, 4, 0, 0, 6, 2, 6, 2, 5, 5, 1, 5, 2, 3, 3, 6, 7, 6, 3, 8, 8, 1, 9, 7, 1, 8, 5, 1, 7, 7, 5, 4, 0, 8, 2, 3, 0, 0, 8, 3, 9, 6, 8, 1, 9, 9, 5, 4, 7, 2, 8, 6, 4, 0, 7, 0, 3

Offset: 0

Wolfdieter Lang, Sep 29 2022

The other (complex) roots are w1*((1 + (1/9)*sqrt(129))/4)^(1/3) + ((1 - (1/9)*sqrt(129))/4)^(1/3) = -0.2119268995... + 1.0652413023...*i, and its complex conjugate, where w1 = (-1 + sqrt(3))/2 = exp((2/3)*Pi*i).

Using hyperbolic functions these roots are -(1/3)*sqrt(3)*(sinh((1/3)*arcsinh((3/4)*sqrt(3))) - sqrt(3)*cosh((1/3)*arcsinh((3/4)*sqrt(3)))*i), and its complex conjugate.

			0.423853799069783271378040062625515233676388197185177540823008396819954728...

Cf. A316711 (Comment).

RealDigits[x /. FindRoot[2*x^3 + 2*x - 1, {x, 1}, WorkingPrecision -> 100]][[1]] (* Amiram Eldar, Sep 29 2022 *)

r = ((1 +(1/9)*sqrt(129))/4)^(1/3) - (1/3)*((1 + (1/9)*sqrt(129))/4)^(-1/3).
r = ((1 + (1/9)*sqrt(129))/4)^(1/3) + w1*((1 - (1/9)*sqrt(129))/4)^(1/3), where w1 = (-1 + sqrt(3))/2, one of the complex roots of x^3 - 1.
r = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/4)*sqrt(3))).

cp's OEIS Frontend

A276383 Complement of A158919: complementary Beatty sequence to the Beatty sequence defined by the tribonacci constant tau = A058265.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A317202 Decimal expansion of 3 + (t^2+t^4)/2, where t = 0.543689... (A192918) is the real root of x^3 + x^2 + x = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A357463 Decimal expansion of the real root of 2x^3 + 2x - 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

cp's OEIS Frontend

A276383 Complement of A158919: complementary Beatty sequence to the Beatty sequence defined by the tribonacci constant tau = A058265.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A317202 Decimal expansion of 3 + (t^2+t^4)/2, where t = 0.543689... (A192918) is the real root of x^3 + x^2 + x = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A357463 Decimal expansion of the real root of 2*x^3 + 2*x - 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A357463 Decimal expansion of the real root of 2x^3 + 2x - 1.