A005480 -id:A005480 - cp's OEIS frontend

A047914 Duplicate of A005480.

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

Offset: 1

dead

A002580 Decimal expansion of cube root of 2.

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

Offset: 1

N. J. A. Sloane

2^(1/3) is Hermite's constant gamma_3. - Jean-François Alcover, Sep 02 2014, after Steven Finch.
For doubling the cube using origami and a standard geometric construction employing two right angles see the W. Lang link, Application 2, p. 14, and the references given there. See also the L. Newton link. - Wolfdieter Lang, Sep 02 2014
Length of an edge of a cube with volume 2. - Jared Kish, Oct 16 2014
For any positive real c, the mappings R(x)=(c*x)^(1/4) and S(x)=sqrt(c/x) have the same unique attractor c^(1/3), to which their iterated applications converge from any complex plane point. The present case is obtained setting c=2. It is noteworthy that in this way one can evaluate cube roots using only square roots. The CROSSREFS list some other cases of cube roots to which this comment might apply. - Stanislav Sykora, Nov 11 2015
The cube root of any positive number can be connected to the Philo lines (or Philon lines) for a 90-degree angle. If the equation x^3-2 is represented using Lill's method, it can be shown that the path of the root 2^(1/3) creates the shortest segment (Philo line) from the x axis through (1,2) to the y axis. For more details see the article "Lill's method and the Philo Line for Right Angles" linked below. - Raul Prisacariu, Apr 06 2024

			1.2599210498948731647672106072782283505702514...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 192-193.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.4 Irrational Numbers and §12.3 Euclidean Construction, pp. 84, 421.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Horace S. Uhler, Many-figure approximations for cubed root of 2, cubed root of 3, cubed root of 4, and cubed root of 9 with chi2 data. Scripta Math. 18, (1952). 173-176.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, pp. 33-34.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Laszlo C. Bardos, Double a Cube, CutOutFoldUp.
Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 62.
Wolfdieter Lang, Notes on Some Geometric and Algebraic Problems solved by Origami, arXiv:1409.4799 [math.MG], 2014.
Liz Newton, The power of origami.
Raul Prisacariu, Lill's method and the Philo Line for Right Angles.
Simon Plouffe, Plouffe's Inverter, The cube root of 2 to 20000 digits.
Simon Plouffe, 2**(1/3) to 2000 places.
Simon Plouffe, Generalized expansion of real constants.
H. S. Uhler, Many-figure approximations for cubed root of 2, cubed root of 3, cubed root of 4, and cubed root of 9 with chi2 data, Scripta Math. 18, (1952). 173-176. [Annotated scanned copies of pages 175 and 176 only]
Eric Weisstein's World of Mathematics, Delian Constant.
Index entries for algebraic numbers, degree 3.

Cf. A002945 (continued fraction), A270714 (reciprocal), A253583.
Cf. A246644.
Cf. A002581, A005480, A005481, A005482, A005486, A010581, A010582, A092039, A092041, A139340.

Digits:=100: evalf(2^(1/3)); # Wesley Ivan Hurt, Nov 12 2015

RealDigits[N[2^(1/3), 5!]] (* Vladimir Joseph Stephan Orlovsky, Sep 04 2008 *)

default(realprecision, 20080); x=2^(1/3); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002580.txt", n, " ", d));  \\ Harry J. Smith, May 07 2009

default(realprecision, 100); x= 2^(1/3); for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", "))  \\ Altug Alkan, Nov 14 2015

(-2^(1/3) - 2^(1/3) * sqrt(-3))^3 = (-2^(1/3) + 2^(1/3) * sqrt(-3))^3 = 16. - Alonso del Arte, Jan 04 2015
Set c=2 in the identities c^(1/3) = sqrt(c/sqrt(c/sqrt(c/...))) = sqrt(sqrt(c*sqrt(sqrt(c*sqrt(sqrt(...)))))). - Stanislav Sykora, Nov 11 2015
Equals Product_{k>=0} (1 + (-1)^k/(3*k + 2)). - Amiram Eldar, Jul 25 2020
From Peter Bala, Mar 01 2022: (Start)
Equals Sum_{n >= 0} (1/(3*n+1) - 1/(3*n-2))*binomial(1/3,n) = (3/2)* hypergeom([-1/3, -2/3], [4/3], -1). Cf. A290570.
Equals 4/3 - 4*Sum_{n >= 1} binomial(1/3,2*n+1)/(6*n-1) = (4/3)*hypergeom ([1/2, -1/6], [3/2], 1).
Equals hypergeom([-2/3, -1/6], [1/2], 1).
Equals hypergeom([2/3, 1/6], [4/3], 1). (End)

A197008 Decimal expansion of the shortest distance from x axis through (1,2) to y axis.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 10 2011

The Philo line of a point P inside an angle T is the shortest segment that crosses T and passes through P.  Suppose that T is the angle formed by the positive x and y axes and that h>0 and k>0.  Notation:
...
P=(h,k)
L=the Philo line of P across T
U=x-intercept of L
V=y-intercept of L
d=|UV|
...
Although Philo lines are not generally Euclidean-constructible, exact expressions for U, V, and d can be found for the angle T under consideration.  Write u(t)=(t,0), let v(t) the corresponding point on the y axis, and let d(t) be the distance between u(t) and v(t). Then d is found by minimizing d(t)^2:
  d=w*sqrt(1+(k/h)^(2/3)), where w=(h+(h*k^2))^(1/3).
...
Guide:
h....k...........d
1....2........A197008
1....3........A197012
1....4........A197013
2....3........A197014
3....4........A197015
1..sqrt(2)....A197031
...
For guides to other Philo lines, see A195284 and A197032.
The cube root of any positive number can be connected to the Philo lines (or Philon lines) for a 90-degree angle. If the equation x^3-2 is represented using Lill's method, it can be shown that the path of the root 2^(1/3) creates the shortest segment (Philo line) from the x axis through (1,2) to the y axis. For more details see the article "Lill's method and the Philo Line for Right Angles" linked below. - Raul Prisacariu, Apr 06 2024

			d=4.161938184941462752390080...
x-intercept: U=(2.5874..., 0)
y-intercept: V=(0, 3.2599...)

R. J. Mathar, OEIS A197008
Raul Prisacariu, Lill's method and the Philo Line for Right Angles.
Index entries for algebraic numbers, degree 6

Cf. A197012, A005480, A195284.

(1+2^(2/3))^(3/2); evalf(%) ; # R. J. Mathar, Nov 08 2022

f[x_] := x^2 + (k*x/(x - h))^2; t = h + (h*k^2)^(1/3);
h = 1; k = 2; d = N[f[t]^(1/2), 100]
RealDigits[d] (* this sequence *)
x = N[t] (* x-intercept; -1+4^(1/3); cf. A005480 *)
y = N[k*t/(t - h)] (* y-intercept *)
Show[Plot[k + k (x - h)/(h - t), {x, 0, t}],
ContourPlot[(x - h)^2 + (y - k)^2 == .001, {x, 0, 4}, {y, 0, 4}], PlotRange -> All, AspectRatio -> Automatic]

polrootsreal(x^6 - 15*x^4 - 33*x^2 - 125)[2] \\ Charles R Greathouse IV, Feb 03 2025

A118292 Decimal expansion of (Gamma(1/6)Gamma(1/3))/(3sqrt(Pi)).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Apr 22 2006

General formula: Integral_{x=0..1} (1+x^(3n))/sqrt(1-x^3) dx = G_3 * k_n = G_3*A146751(n)/A146752(n) = A118292*A146751(n)/A146752(n) where G_3 = (Gamma(1/3)^3)/(2^(1/3)*sqrt(3)*Pi) is the number in the present entry. For numerators of k_n see A146752, for denominators of k_n see A146753. - Artur Jasinski

gamma(1/6)*gamma(1/3)/(3*sqrt(Pi)) = gamma(1/3)^3/(2^(1/3)*sqrt(3)*Pi). - Harry J. Smith, May 09 2009

			2.8043642106509085223500381581005882709260444108....

Harry J. Smith, Table of n, a(n) for n = 1..4000
Eric Weisstein's World of Mathematics, Butterfly Curve

Cf. A146752, A146753, A160323 (continued fraction).

RealDigits[(Gamma[1/3]^3)/(2^(1/3) Sqrt[3] Pi), 10, 200] (* Artur Jasinski*)

allocatemem(932245000); default(realprecision, 4080); x=gamma(1/3)^3/(2^(1/3)*sqrt(3)*Pi); for (n=1, 4000, d=floor(x); x=(x-d)*10; write("b118292.txt", n, " ", d)) \\ Harry J. Smith, Jun 20 2009

3/hypergeom([1/3,1/6],[3/2],1) \\ Charles R Greathouse IV, Aug 29 2025

Equals A073005^3 / (A002194*A002580*A000796) [see Vidunas, arXiv:math.CA/0403510]. - R. J. Mathar, Nov 30 2008

Equals 3/hypergeom([1/3, 1/6], [3/2], 1) = A290570*A005480. - Peter Bala, Mar 02 2022

Edited by N. J. A. Sloane, Nov 16 2008 at the suggestion of R. J. Mathar

A010769 Decimal expansion of 7th root of 2.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

This is also the unique positive attractor of the mapping M(x) = sqrt(sqrt(sqrt(2*x))). In general, (p^N-1)-th root of a number f can be approximated by iterating the mapping M(x) = (f*x)^(1/p^N). The convergence is very fast. In this case, p=2, N=3, and f=2. In the form "evaluate the 3rd (or 7th or 15th) root of a number using only square roots", the insight is usable as a recreational math puzzle. - Stanislav Sykora, Oct 26 2015

			1.104089513673812337649505387623...

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cube roots (p=2,N=2) for various f: A002580 (2), A002581 (3), A005480 (4), A010582 (10), A092041 (e). 7th roots (p=2,N=3): A246709 (3), A011186 (4), A011201 (5), A011276 (10), A092516 (e). 8th roots (p=3,N=2): A010770 (2), A246710 (3), A011202 (5), A011277 (10). 15th roots (p=2,N=4): A010777(2), A011194(4), A011209(5), A011284(10). - Stanislav Sykora, Oct 26 2015

RealDigits[N[2^(1/7), 100]][[1]] (* Vincenzo Librandi, Apr 02 2013 *)
RealDigits[Surd[2,7],10,120][[1]] (* Harvey P. Dale, Sep 05 2022 *)

sqrtn(2,7) \\ Charles R Greathouse IV, Apr 15 2014

{ default(realprecision, 100); x= 2^(1/7); for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", ")) } \\ Altug Alkan, Nov 14 2015

Equals Product_{k>=0} (1 + (-1)^k/(7*k + 6)). - Amiram Eldar, Jul 29 2020

A182773 Beatty sequence for 1+2^(2/3).

Original entry on oeis.org

2, 5, 7, 10, 12, 15, 18, 20, 23, 25, 28, 31, 33, 36, 38, 41, 43, 46, 49, 51, 54, 56, 59, 62, 64, 67, 69, 72, 75, 77, 80, 82, 85, 87, 90, 93, 95, 98, 100, 103, 106, 108, 111, 113, 116, 119, 121, 124, 126, 129, 131, 134, 137, 139, 142

Offset: 1

Clark Kimberling, Nov 30 2010

nonn

Let u=2^(1/3). Jointly rank {j*u} and {k/u} as in the first comment at A182760; a(n) is the position of n*u.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences.

Cf. A005480, A182760, A182774.

Floor[Range[100]*(1 + 2^(2/3))] (* Paolo Xausa, Jul 09 2024 *)

a(n) = floor(n*(1+2^(2/3))).

A235362 Decimal expansion of the cube root of 2 divided by 2.

Original entry on oeis.org

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

Offset: 0

Alonso del Arte, Jan 07 2014

Also reciprocal of the real cubic root of 4 and negated real part of either complex cubic root of 2.

			0.6299605249474365823836053...

Richard Zippel, The Weyl Computer Algebra Substrate in Design and Implementation of Symbolic Computation Systems: International Symposium, DISCO '93 Gmunden, Austria, September 15-17, 1993. Proceedings, p. 306.

Cf. A002580, A010503, A020761, A239797.

Digits := 100 ; evalf(1/2^(2/3)) ; # R. J. Mathar, Jan 16 2023

Mathematica
```
RealDigits[1/2^(2/3), 10, 128][[1]]
```

sqrtn(1/4,3) \\ Charles R Greathouse IV, Apr 14 2014

2^(1/3)/2 = 1/2^(2/3) = 1/4^(1/3).
(-2^(1/3)/2 + sqrt(-3)/4^(1/3))^3 = 2.
Equals 1/A005480 = A002580 /2 . - Wolfdieter Lang, Jan 02 2023

A270714 Decimal expansion of (1/2)^(1/3).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Mar 22 2016

Let c = (1/2)^(1/3). A sphere of radius c*r has half the volume of a sphere of radius r. - Rick L. Shepherd, Aug 12 2016

Let c = (1/2)^(1/3). The relative maximum of xy(x+y)=1 is (c,-1/c^2). - Clark Kimberling, Oct 05 2020

			0.79370052598409973737585281963615413019574666394992650490414288091260825...

Clark Kimberling, Table of n, a(n) for n = 0..10000

Cf. A002580, A005480, A235362, A269993.

Mathematica
```
RealDigits[(1/2)^(1/3), 10, 200][[1]]
```
PARI
```
(1/2)^(1/3) \\ Altug Alkan, Mar 22 2016
```

Equals 1/A002580 = A002580*A235362 = A005480*A020761. [corrected and expanded by Rick L. Shepherd, Aug 12 2016]

Equals Product_{k>=1} (1 + (-1)^k/(3*k+1)). - Amiram Eldar, Aug 10 2020

A002355 Denominators of convergents to cube root of 4.

Original entry on oeis.org

1, 1, 2, 5, 12, 17, 63, 143, 492, 635, 2397, 3032, 93357, 96389, 478913, 575302, 1629517, 15240955, 93075247, 387541943, 480617190, 868159133, 2216935456, 16386707325, 34990350106, 121357757643, 277705865392, 399063623035, 2672087603602, 3071151226637

Offset: 0

N. J. A. Sloane, Herman P. Robinson

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 67.
P. Seeling, Verwandlung der irrationalen Groesse ... in einen Kettenbruch, Archiv. Math. Phys., 46 (1866), 80-120.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..999

Cf. A002356 (numerators), A005480.

Denominator[Convergents[Power[4, (3)^-1], 30]] (* Vincenzo Librandi, Aug 24 2013 *)

a(n) = contfracpnqn(contfrac(4^(1/3), n))[2, 1]; \\ Michel Marcus, Aug 23 2013

More terms from Vincenzo Librandi, Aug 24 2013

Offset changed by Andrew Howroyd, Jul 04 2024

A002356 Numerators of convergents to cube root of 4.

Original entry on oeis.org

1, 2, 3, 8, 19, 27, 100, 227, 781, 1008, 3805, 4813, 148195, 153008, 760227, 913235, 2586697, 24193508, 147747745, 615184488, 762932233, 1378116721, 3519165675, 26012276446, 55543718567, 192643432147, 440830582861, 633474015008, 4241674672909, 4875148687917

Offset: 0

N. J. A. Sloane

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 67.
P. Seeling, Verwandlung der irrationalen Groesse ... in einen Kettenbruch, Archiv. Math. Phys., 46 (1866), 80-120.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..999

Cf. A002355 (denominators), A005480.

Numerator[Convergents[Power[4, (3)^-1], 30]] (* Vincenzo Librandi, Aug 24 2013 *)

a(n) = contfracpnqn(contfrac(4^(1/3), n))[1][1]; \\ Michel Marcus, Aug 23 2013

More terms from Herman P. Robinson
More terms from Vincenzo Librandi, Aug 24 2013
Offset changed by Andrew Howroyd, Jul 04 2024

cp's OEIS Frontend

A047914 Duplicate of A005480.

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A002580 Decimal expansion of cube root of 2.

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A197008 Decimal expansion of the shortest distance from x axis through (1,2) to y axis.

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A118292 Decimal expansion of (Gamma(1/6)*Gamma(1/3))/(3*sqrt(Pi)).

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A010769 Decimal expansion of 7th root of 2.

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A182773 Beatty sequence for 1+2^(2/3).

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A235362 Decimal expansion of the cube root of 2 divided by 2.

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A118292 Decimal expansion of (Gamma(1/6)Gamma(1/3))/(3sqrt(Pi)).