A228497 -id:A228497 - cp's OEIS frontend

A010767 Decimal expansion of 4th root of 2.

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

Offset: 1

N. J. A. Sloane

An algebraic integer of degree 4. - Charles R Greathouse IV, Nov 12 2014

			1.189207115002721066717499970560475915292972092...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.23, p. 407.

A.H.M. Smeets, Table of n, a(n) for n = 1..20001 (first 1000 digits from Vincenzo Librandi).
Jean-Paul Allouche, Henri Cohen, Michel Mendès France, and Jeffrey O. Shallit, De nouveaux curieux produits infinis, Acta Arithmetica, Vol. 49, No. 2 (1987), pp. 141-153; alternative link.
Simon Plouffe, 2^(1/4) or sqrt(sqrt(2)) to 20000 digits.
Simon Plouffe, 2^(1/4) to 1024 places.
Nikita Sidorov and Boris Solomyak, On the topology of sums in powers of an algebraic number, arXiv:0909.3324 [math.NT], 2009-2011.
David Terr and Eric W. Weisstein, Pisot Number.
Eric Weisstein's World of Mathematics, Algebraic integer.
Index entries for algebraic numbers, degree 4.

Cf. A000120, A228497 (the multiplicative inverse).

RealDigits[N[2^(1/4),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jan 24 2012 *)

sqrtn(2,4) \\ Charles R Greathouse IV, Apr 14 2014

weber(I) \\ Charles R Greathouse IV, Feb 04 2015

Equals Product_{k>=0} (1 + (-1)^k/(4*k + 3)). - Amiram Eldar, Jul 25 2020

Equals Product_{k>=0} ((2*k+1)/(2*k+2))^(A000120(k)*(-1)^A000120(k)) (Allouche et al., 1987). - Amiram Eldar, Feb 04 2024

A154747 Decimal expansion of sqrt(sqrt(2) - 1), the radius vector of the point of bisection of the arc of the unit lemniscate (x^2 + y^2)^2 = x^2 - y^2 in the first quadrant.

Original entry on oeis.org

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

Offset: 0

Stuart Clary, Jan 14 2009

A root of r^4 + 2 r^2 - 1 = 0.
Also real part of sqrt(1 + i)^3, where i is the imaginary unit such that i^2 = -1. - Alonso del Arte, Sep 09 2019
From Bernard Schott, Dec 19 2020: (Start)
Length of the shortest line segment which divides a right isosceles triangle with AB = AC = 1 into two parts of equal area; this is the answer to the 2nd problem proposed during the final round of the 18th British Mathematical Olympiad in 1993 (see link BM0 and Gardiner reference).
The length of this shortest line segment IJ with I on a short side and J on the hypotenuse is sqrt(sqrt(2)-1), and AI = AJ = 1/sqrt(sqrt(2)) = A228497 (see link Figure for B.M.O. 1993, Problem 2). (End)
This algebraic number and its negation equal the real roots of the quartic x^4 + 2*x^2 - 1 (minimal polynomial). The imaginary roots are +A278928*i and -A278928*i. - Wolfdieter Lang, Sep 23 2022

			0.643594252905582624735443437418...

A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Problem 2, pages 56 and 104-105 (1993).
C. L. Siegel, Topics in Complex Function Theory, Volume I: Elliptic Functions and Uniformization Theory, Wiley-Interscience, 1969, page 5

G. C. Greubel, Table of n, a(n) for n = 0..5000.
British Mathematical Olympiad 1993, Problem 2.
Bernard Schott, Figure for B.M.O. 1993, Problem 2.
Index entries for algebraic numbers, degree 4
Index to sequences related to curves.
Index to sequences related to Olympiads.

Cf. A154739 for the abscissa and A154743 for the ordinate.
Cf. A154748, A154749 and A154750 for the continued fraction and the numerators and denominators of the convergents.
Cf. A085565 for 1.311028777..., the first-quadrant arc length of the unit lemniscate.
Cf. A309948 and A309949 for real and imaginary parts of sqrt(1 + i).
Cf. A278928.

nmax = 1000; First[ RealDigits[ Sqrt[Sqrt[2] - 1], 10, nmax] ]

sqrt(sqrt(2) - 1) \\ Michel Marcus, Dec 10 2016

From Peter Bala, Jul 01 2024: (Start)
This constant occurs in the evaluation of Integral_{x = 0..Pi/2} sin^2(x)/(1 + sin^4(x)) dx = Pi/4 * sqrt(sqrt(2) - 1).
Equals (1/2)*Sum_{n >= 0} (-1/16)^n * binomial(4*n+2, 2*n+1) (a slowly converging series). (End)
Equals 2^(3/4)*sin(Pi/8). - Vaclav Kotesovec, Jul 01 2024

Offset corrected by R. J. Mathar, Feb 05 2009

A228496 Decimal expansion of arccos(2/3).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 23 2013

The value describes the smaller of the internal angles in the triangle of the surfaces of the Tetrakis Hexahedron.
The value equals Pi/2 minus A156547, the 90-degree complement to 41.81031... degrees.
The value is a little bit larger than the 4th root of 1/2, which is 0.8408964... = A228497.
If a ball assimilated to a point rolls without friction on a sphere starting from the top with zero initial velocity, this value is the angle in radians, measured at the center of the sphere, from the top of the sphere to the point at which the ball leaves the surface of the sphere. See Jayanth et al. - Robert FERREOL, Sep 14 2019
The maximum possible value of the least of the nine acute angles between pairs of edges of two randomly disoriented cubes (in radians, see A361618). - Amiram Eldar, Mar 18 2023

			Equals 0.8410686705679302557765... radians  = 48.189685... degrees.

emc2, Bille glissant sur une sphère (in French).
V. Jayanth, C. Raghunandan and Anindya Kumar Biswas, A sphere moving down the surface of a static sphere and a simple phase diagram, arXiv:0808.3531 [physics.class-ph], 2008-2009.
R. J. Mathar, Hierarchical Subdivision of the Simple Cubic Lattice, arXiv preprint arXiv:1309.3705 [math.MG], 2013.
Wikipedia, Tetrakis hexahedron.
Index entries for transcendental numbers

Cf. A156547, A228497, A361618.

Maple
```
evalf(arccos(2/3)) ;
```

RealDigits[ArcCos[2/3], 10, 100][[1]] (* Amiram Eldar, May 24 2021 *)

PARI
```
acos(2/3) \\ Michel Marcus, Sep 14 2019
```

Equals arcsin(sqrt(5)/3).

Equals arctan(sqrt(5)/2). - Amiram Eldar, May 24 2021

A011006 Decimal expansion of 4th root of 8.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			1.68179283...

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for algebraic numbers, degree 4.

RealDigits[N[8^(1/4), 100]][[1]] (* Vincenzo Librandi, Apr 04 2013 *)

sqrtn(8,4) \\ Charles R Greathouse IV, Apr 15 2014

Equals 2*A228497 =2^(3/4) = sqrt(A010466). - R. J. Mathar, Jan 15 2021

A308320 Decimal expansion of 2^(-7/4); exact length of the A4 paper size measured in meters according to the ISO 216 standard.

Original entry on oeis.org

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

Offset: 0

Jianing Song, May 20 2019

Also exact width of the A3 paper size measured in meters.

According to the ISO 216 standard, the A0 paper size is defined to have an area of 1 square meter where the ratio of the length to the width is sqrt(2), so the length is 2^(1/4) m and the width is 2^(-1/4) m. For each n >= 0, the length of the size A(n+1) is equal to the width of the size A(n) and the width of the size A(n+1) is equal to half of the length of the size A(n), so the area of the size A(n+1) is half of that of A(n). Equivalently, the length of the A(n) size is 2^(-n/2 + 1/4) m and the width is 2^(-n/2 - 1/4) m. For the A4 size, the exact length and width are 2^(-7/4) m = 297.301... mm and 2^(-9/4) m = 210.224... mm (A308321), and the actual length and width are 297 mm and 210 mm.

			0.29730177...
The exact lengths and widths (rounded to the nearest 1/10 mm) and areas of the A-series are as follows:
.
  size |       exact length      |       exact width      | exact area (mm^2)
   A0  | 2^(  1/4) m = 1189.2 mm | 2^(- 1/4) m = 840.9 mm |  1000000
   A1  | 2^(- 1/4) m =  840.9 mm | 2^(- 3/4) m = 594.6 mm |   500000
   A2  | 2^(- 3/4) m =  594.6 mm | 2^(- 5/4) m = 420.4 mm |   250000
   A3  | 2^(- 5/4) m =  420.4 mm | 2^(- 7/4) m = 297.3 mm |   125000
   A4  | 2^(- 7/4) m =  297.3 mm | 2^(- 9/4) m = 210.2 mm |    62500
   A5  | 2^(- 9/4) m =  210.2 mm | 2^(-11/4) m = 148.7 mm |    31250
   A6  | 2^(-11/4) m =  148.7 mm | 2^(-13/4) m = 105.1 mm |    15625
   A7  | 2^(-13/4) m =  105.1 mm | 2^(-15/4) m =  74.3 mm |     7812.5
   A8  | 2^(-15/4) m =   74.3 mm | 2^(-17/4) m =  52.6 mm |     3906.25
   A9  | 2^(-17/4) m =   52.6 mm | 2^(-19/4) m =  37.2 mm |     1953.125
   A10 | 2^(-19/4) m =   37.2 mm | 2^(-21/4) m =  26.3 mm |      976.5625
.
And the actual lengths, widths and areas (note that the actual areas are always smaller than the exact areas) are as follows:
.
  size | actual length (mm) | actual width (mm) | actual area (mm^2)
   A0  |        1189        |        841        |  999949 (99.9949%)
   A1  |         841        |        594        |  499554 (99.9108%)
   A2  |         594        |        420        |  249480 (99.7920%)
   A3  |         420        |        297        |  124740 (99.7920%)
   A4  |         297        |        210        |   62370 (99.7920%)
   A5  |         210        |        148        |   31080 (99.4560%)
   A6  |         148        |        105        |   15540 (99.4560%)
   A7  |         105        |         74        |    7770 (99.4560%)
   A8  |          74        |         52        |    3848 (98.5088%)
   A9  |          52        |         37        |    1924 (98.5088%)
   A10 |          37        |         26        |     962 (98.5088%)

Prepressure, A4.
Wikipedia, ISO 216.

Cf. A010767 (2^(1/4)), A228497 (2^(-1/4)), A308321 (2^(-9/4)).

RealDigits[2^(-7/4),10,88][[1]] (* James C. McMahon, Feb 26 2024 *)

PARI
```
default(realprecision, 100); 2^(-7/4)
```

Equals square root of A222066. - R. J. Mathar, Jan 27 2021

Edited by Jon E. Schoenfield, Feb 25 2024

A308321 Decimal expansion of 2^(-9/4); exact width of the A4 paper size measured in meters according to the ISO 216 standard.

Original entry on oeis.org

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

Offset: 0

Jianing Song, May 20 2019

Also exact length of the A5 paper size measured in meters.

According to the ISO 216 standard, the A0 paper size is defined to have an area of 1 square meter where the ratio of the length to the width is sqrt(2), so the length is 2^(1/4) m and the width is 2^(-1/4) m. For each n >= 0, the length of the size A(n+1) is equal to the width of the size A(n) and the width of the size A(n+1) is equal to half of the length of the size A(n), so the area of the size A(n+1) is half of that of A(n). Equivalently, the length of the A(n) size is 2^(-n/2 + 1/4) m and the width is 2^(-n/2 - 1/4) m. For the A4 size, the exact length and width are 2^(-7/4) m = 297.301... mm (A308320) and 2^(-9/4) m = 210.224... mm, and the actual length and width are 297 mm and 210 mm.

			The exact lengths and widths (rounded to the nearest 1/10 mm) and areas of the A-series are as follows:
.
  size |       exact length      |       exact width      | exact area (mm^2)
   A0  | 2^(  1/4) m = 1189.2 mm | 2^(- 1/4) m = 840.9 mm |  1000000
   A1  | 2^(- 1/4) m =  840.9 mm | 2^(- 3/4) m = 594.6 mm |   500000
   A2  | 2^(- 3/4) m =  594.6 mm | 2^(- 5/4) m = 420.4 mm |   250000
   A3  | 2^(- 5/4) m =  420.4 mm | 2^(- 7/4) m = 297.3 mm |   125000
   A4  | 2^(- 7/4) m =  297.3 mm | 2^(- 9/4) m = 210.2 mm |    62500
   A5  | 2^(- 9/4) m =  210.2 mm | 2^(-11/4) m = 148.7 mm |    31250
   A6  | 2^(-11/4) m =  148.7 mm | 2^(-13/4) m = 105.1 mm |    15625
   A7  | 2^(-13/4) m =  105.1 mm | 2^(-15/4) m =  74.3 mm |     7812.5
   A8  | 2^(-15/4) m =   74.3 mm | 2^(-17/4) m =  52.6 mm |     3906.25
   A9  | 2^(-17/4) m =   52.6 mm | 2^(-19/4) m =  37.2 mm |     1953.125
   A10 | 2^(-19/4) m =   37.2 mm | 2^(-21/4) m =  26.3 mm |      976.5625
.
And the actual lengths, widths and areas (note that the actual areas are always smaller than the exact areas) are as follows:
.
  size | actual length (mm) | actual width (mm) | actual area (mm^2)
   A0  |        1189        |        841        |  999949 (99.9949%)
   A1  |         841        |        594        |  499554 (99.9108%)
   A2  |         594        |        420        |  249480 (99.7920%)
   A3  |         420        |        297        |  124740 (99.7920%)
   A4  |         297        |        210        |   62370 (99.7920%)
   A5  |         210        |        148        |   31080 (99.4560%)
   A6  |         148        |        105        |   15540 (99.4560%)
   A7  |         105        |         74        |    7770 (99.4560%)
   A8  |          74        |         52        |    3848 (98.5088%)
   A9  |          52        |         37        |    1924 (98.5088%)
   A10 |          37        |         26        |     962 (98.5088%)

Prepressure, A4.
Wikipedia, ISO 216.

Cf. A010767 (2^(1/4)), A228497 (2^(-1/4)), A308320 (2^(-7/4)).

RealDigits[2^(-9/4),10,88][[1]] (* James C. McMahon, Feb 26 2024 *)

PARI
```
default(realprecision, 100); 2^(-9/4)
```

Edited by Jon E. Schoenfield, Feb 25 2024

A380907 Decimal expansion of 1/(2^(1/4)*sqrt(1+Pi/4)).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Feb 08 2025

			0.62932496342101931026228634377882172549266644...

Piotr Garbaczewski and Vladimir A. Stephanovich, Dynamics of Confined Lévy Flights in Terms of (Lévy) Semigroups, Acta Phys. Pol. B 43, 977-999 (2012). See p. 992.

Cf. A003881, A010767, A019704, A228497.

RealDigits[1/(2^(1/4)Sqrt[1+Pi/4]),10,100][[1]]

cp's OEIS Frontend

A010767 Decimal expansion of 4th root of 2.

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A154747 Decimal expansion of sqrt(sqrt(2) - 1), the radius vector of the point of bisection of the arc of the unit lemniscate (x^2 + y^2)^2 = x^2 - y^2 in the first quadrant.

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A228496 Decimal expansion of arccos(2/3).

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A011006 Decimal expansion of 4th root of 8.

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A308320 Decimal expansion of 2^(-7/4); exact length of the A4 paper size measured in meters according to the ISO 216 standard.

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A308321 Decimal expansion of 2^(-9/4); exact width of the A4 paper size measured in meters according to the ISO 216 standard.

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A380907 Decimal expansion of 1/(2^(1/4)*sqrt(1+Pi/4)).

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