A182168 -id:A182168 - cp's OEIS frontend

A232736 Decimal expansion of sin(Pi/14), or the imaginary part of (-1)^(1/7).

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

Offset: 0

Stanislav Sykora, Nov 29 2013

The corresponding real part is in A232735.
Root of the equation 1 - 4*x - 4*x^2 + 8*x^3 = 0. - Vaclav Kotesovec, Apr 04 2021
The other 2 roots are -A362922 and A073052. - R. J. Mathar, Aug 29 2025

			0.222520933956314404288902564496794759466355568764544955311987...

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 3.

Cf. A232735 (real part), A010503 (imag(I^(1/2))), A182168 (imag(I^(1/4))), A019827 (imag(I^(1/5))), A019824 (imag(I^(1/6))), A232738 (imag(I^(1/8))), A019819 (imag(I^(1/9))), A019818 (imag(I^(1/10))).

A101464 Decimal expansion of sqrt(2-sqrt(2)), edge length of a regular octagon with circumradius 1.

Original entry on oeis.org

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

Offset: 0

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Jan 20 2005

			0.765366864730179543456919968060797733522689124971254082867601271255092067920...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Eric Weisstein's World of Mathematics, Octagon.
Index entries for algebraic numbers, degree 4.

Cf. A047621, A101465, A179260 (sqrt(2+sqrt(2))), A182168, A285871, A329246.

RealDigits[Sqrt[2-Sqrt[2]],10,120][[1]] (* Harvey P. Dale, Jun 22 2011 *)

2*sin(Pi/8) \\ Charles R Greathouse IV, Feb 04 2025

polrootsreal(x^4-4*x^2+2)[3] \\ Charles R Greathouse IV, Feb 04 2025

Equals i^(3/4) + i^(-3/4). - Gary W. Adamson, Jul 07 2022
Equals 2*sin(Pi/8) = 2*A182168. - Amiram Eldar, Apr 06 2023
Equals Product_ {k >= 0} ((8*k - 2)*(8*k + 10))/((8*k - 5)*(8*k + 13)). - Antonio Graciá Llorente, Mar 11 2024
Equals Product_{k>=1} (1 + (-1)^k/A047621(k)). - Amiram Eldar, Nov 22 2024
Equals sqrt(A101465) = 1/A285871 = exp(-A329246). - Hugo Pfoertner, Nov 22 2024

A154739 Decimal expansion of sqrt(1 - 1/sqrt(2)), the abscissa 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

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

Offset: 0

Stuart Clary, Jan 14 2009

A root of 2*x^4 - 4*x^2 + 1 = 0.

			0.541196100146196984399723205366...

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
Index entries for algebraic numbers, degree 4.

Cf. A154743 for the ordinate and A154747 for the radius vector.
Cf. A154740, A154741 and A154742 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. A002193, A016825, A179260, A182168, A268682.

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

sqrt(1 - 1/sqrt(2)) \\ G. C. Greubel, Sep 23 2017

polrootsreal(2*x^4-4*x^2+1)[3] \\ Charles R Greathouse IV, Feb 04 2025

From Amiram Eldar, Nov 22 2024: (Start)
Equals sqrt(2) * sin(Pi/8) = A002193 * A182168.
Equals Product_{k>=0} (1 - (-1)^k/(4*k+2)) = Product_{k>=1} (1 + (-1)^k/A016825(k)). (End)
Equals 1/A179260 = sqrt(A268682). - Hugo Pfoertner, Nov 22 2024

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

A188582 Decimal expansion of sqrt(2) - 1.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Apr 04 2011

"In his Book 'The Theory of Poker,' David Sklansky coined the phrase 'Fundamental Theorem of Poker,' a tongue-in-cheek reference to the Fundamental Theorem of Algebra and Fundamental Theorem of Calculus from introductory texts on those two subjects. The constant [sqrt(2) - 1] appears so often in poker analysis that we will in the same vein go so far as to call it 'the golden mean of poker,' and we call it 'r' for short. We will see this value in a number of important results throughout this book." [Chen and Ankenman]
If a triangle has sides whose lengths form a harmonic progression in the ratio 1/(1 - d) : 1 : 1/(1 + d) then the triangle inequality condition requires that d be in the range 1 - sqrt(2) < d < sqrt(2) - 1. - Frank M Jackson, Oct 01 2013
This constant is the 6th smallest radius r < 1 for which a compact packing of the plane exists, with disks of radius 1 and r. - Jean-François Alcover, Sep 02 2014, after Steven Finch
This constant is also the largest argument of the arctangent function in the Viète-like formula for Pi given by Pi/2^(k+1) = arctan(sqrt(2 - a_(k-1))/a_k), where the index k >= 2 and the nested radicals are defined by recurrence using the relations a_k = sqrt(2 + a_(k-1)), a_1 = sqrt(2). When k = 2 the argument of the arctangent function sqrt(2 - a_1)/a_2 = sqrt(2 - sqrt(2))/sqrt(2 + sqrt(2)) = sqrt(2) - 1 is largest. Consequently, at k = 2 the Viète-like formula for Pi can be written as Pi/8 = arctan(sqrt(2 - sqrt(2))/sqrt(2 + sqrt(2))) = arctan(sqrt(2) - 1) (after Abrarov-Quine, see the article). - Sanjar Abrarov, Jan 07 2017
If r and R are respectively the inradius and the circumradius of a triangle, then the ratio r/R <= 1/2 (Euler inequality), and this maximum value 1/2 is obtained when the triangle is equilateral. Now, for a right triangle, the ratio r/R <= this constant = sqrt(2) - 1, and this maximum value sqrt(2) - 1 is obtained when the right triangle is isosceles. This is the answer to the question 1 of the Olympiade Mathématique Belge Maxi in 2008. - Bernard Schott, Sep 07 2022

			0.414213562373095048801688724209698078569671875376948073...

B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, p. 140, Entry 25.
Bill Chen and Jerrod Ankenman, The Mathematics of Poker, Chpt 14 - You Don't Have To Guess: No-Limit Bet Sizing, p. 153, ConJelCo, LLC, Pittsburgh PA 2006.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, pp. 396 and 486.

G. C. Greubel, Table of n, a(n) for n = 0..10000
S. M. Abrarov and B. M. Quine, A Viète-like formula for pi based on infinite sum of the arctangent functions with nested radicals, figshare, 4509014, (2017).
Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 62.
Olympiade Mathématique Belge, OMB 2008, Finale Maxi, Question 1.
Index entries for algebraic numbers, degree 2
Index to sequences related to Olympiads.

Cf. A002193, A014176, A020807, A120731, A182168 (sin(Pi/8)), A144981 (cos(Pi/8)).

Sqrt(2) - 1; // G. C. Greubel, Jan 31 2018

Mathematica
```
RealDigits[ Sqrt[2] - 1, 10, 111][[1]]
```

sqrt(2) - 1 \\ G. C. Greubel, Jan 31 2018

Equals exp(asinh(cos(Pi))) = exp(asinh(-1)). - Geoffrey Caveney, Apr 23 2014
Equals tan(Pi/8) = A182168 / A144981 = 1 / A014176. - Bernard Schott, Apr 12 2022
From Antonio Graciá Llorente, Mar 15 2024: (Start)
Equals Product_{k >= 0} ((8*k - 1)*(8*k + 9))/((8*k - 5)*(8*k + 13)).
Equals Product_{k >= 1} A047554(k)/A047447(k). (End)
From  Peter Bala, Mar 24 2024: (Start)
An infinite family of continued fraction expansions for this constant can be obtained from Berndt, Entry 25, by setting n = 1/2 and x = 8*k + 2 for k >= 0.
For example, taking k = 0 and k = 1 yields
Equals 1/(2 + (1*3)/(4 + (5*7)/(4 + (9*11)/(4 + (13*15)/(4 + ... + (4*n + 1)*(4*n + 3)/(4 + ...)))))) and
Equals (21/5) * 1/(10 + (1*3)/(20 + (5*7)/(20 + (9*11)/(20 + (13*15)/(20 + ... + (4*n + 1)*(4*n + 3)/(20 + ...)))))). (End)
Tan(arctan(c) + arctan(c^3)) = 1/2. - Gary W. Adamson, Apr 04 2024

A232738 Decimal expansion of the imaginary part of I^(1/8), or sin(Pi/16).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 29 2013

The corresponding real part is in A232737.

			0.195090322016128267848284868477022240927691617751954807754502...

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 8

Cf. A232737 (real part), A010503 (imag(I^(1/2))), A182168 (imag(I^(1/4))), A019827 (imag(I^(1/5))), A019824 (imag(I^(1/6))), A232736 (imag(I^(1/7))), A019819 (imag(I^(1/9))), A019818 (imag(I^(1/10))).

R:= RealField(116); Sin(Pi(R)/16); // G. C. Greubel, Sep 20 2022

RealDigits[Sin[Pi/16],10,120][[1]] (* Harvey P. Dale, Sep 01 2018 *)

imag(I^(1/8)) \\ Seiichi Manyama, Apr 04 2021

sin(Pi/16) \\ Seiichi Manyama, Apr 04 2021

sqrt(2-sqrt(2+sqrt(2)))/2 \\ Seiichi Manyama, Apr 04 2021

numerical_approx(sin(pi/16), digits=116) # G. C. Greubel, Sep 20 2022

Equals (1/2) * sqrt(2-sqrt(2+sqrt(2))). - Seiichi Manyama, Apr 04 2021
This^2 + A232737^2 = 1.
Smallest positive of the 8 real-valued roots of 128*x^8-256*x^6+160*x^4-32*x^2+1=0.

A280585 Decimal expansion of 8*sin(Pi/8).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jan 05 2017

Decimal expansion of the ratio of the perimeter of a regular 8-gon (octagon) to its diameter (largest diagonal).

			3.061467458920718173827679872243190934090756499885016331470405085020368271...

Index entries for algebraic numbers, degree 4.

Cf. For other n-gons: A010466 (n=4), 10*A019827 (n=5, 10), A280533 (n=7), A280633 (n=9), A280725 (n=11), A280819 (n=12).

Cf. A182168.

evalf(8*sin(Pi/8),100); # Wesley Ivan Hurt, Feb 01 2017

RealDigits[8*Sin[Pi/8], 10, 120][[1]] (* Amiram Eldar, Jun 26 2023 *)

PARI
```
8*sin(Pi/8)
```

Equals 8*A182168.

A309949 Decimal expansion of the imaginary part of the square root of 1 + i.

Original entry on oeis.org

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

Offset: 0

Alonso del Arte, Aug 24 2019

i is the imaginary unit such that i^2 = -1.

Multiplied by -1, this is the imaginary part of the square root of 1 - i. And also the real part of -sqrt(1 + i) - i + sqrt(1 + i)^3, which is a unit in Q(sqrt(1 + i)).

			Im(sqrt(1 + i)) = 0.45508986056222734130435775782247...

Cf. A000108, A010767, A182168, A309948 (real part).

Digits := 120: Re(sqrt(-1 - I))*10^95:
ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Sep 20 2019

RealDigits[Sqrt[1/Sqrt[2] - 1/2], 10, 100][[1]]

imag(sqrt(1+I)) \\ Michel Marcus, Sep 16 2019

Equals sqrt(1/sqrt(2) - 1/2) = 2^(1/4) * sin(Pi/8).
Equals sqrt((sqrt(2) - 1)/2) = A010767 * A182168. - Bernard Schott, Sep 16 2019
Equals Re(sqrt(-1 - i)). - Peter Luschny, Sep 20 2019
Equals Product_{k>=0} ((8*k - 1)*(8*k + 4))/((8*k - 2)*(8*k + 5)). - Antonio Graciá Llorente, Feb 24 2024

A386241 Decimal expansion of sqrt(5)*sin(Pi/8).

Original entry on oeis.org

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

Offset: 0

Hugo Pfoertner, Jul 18 2025

Upper bound of the wobbling distance S of two rotated square lattices. See A307110 and A307731 for the special case of rotation angle Pi/4. According to Jan Fricke (1999), the angle Pi/4 is the most unfavorable case, i.e., smaller bounds can be found for all other angles.

			0.8557061686312838477748180718246837073...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Hans-Georg Carstens, Walter A. Deuber, Wolfgang Thumser, and Elke Koppenrade, Geometrical Bijections in Discrete Lattices. Combinatorics, Probability and Computing, 8(1-2), 109-129, 1999.
Jan Fricke, Symmetric Graphs have symmetric Matchings, arXiv:1607.07426 [math.GR], 25 Jul 2016, Introduction.
Klaus Nagel, A Lower Bound for Double Lattice Bijections, August 17, 2025.
Index entries for algebraic numbers, degree 4

Cf. A002163, A144981, A182168, A307110, A307731, A367150.

First[RealDigits[Sqrt[5]*Sin[Pi/8],10,100]] (* Paolo Xausa, Aug 13 2025 *)

sqrt(5)*sin(Pi/8) \\ Charles R Greathouse IV, Aug 19 2025

Equals A002163*A182168.

The minimal polynomial is 8*x^4 - 40*x^2 + 25. - Joerg Arndt, Aug 02 2025

A343055 Decimal expansion of the imaginary part of i^(1/16), or sin(Pi/32).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

An algebraic number of degree 16 and denominator 2. - Charles R Greathouse IV, Jan 09 2022

			0.09801714032956060199419...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Penn, The exact value of sin 5⅝°??, YouTube video, 2021.
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 16

sin(Pi/m): A010527 (m=3), A010503 (m=4), A019845 (m=5), A323601 (m=7), A182168 (m=8), A019829 (m=9), A019827 (m=10), A019824 (m=12), A232736 (m=14), A019821 (m=15), A232738 (m=16), A241243 (m=17), A019819 (m=18), A019818 (m=20), A343054 (m=24), A019815 (m=30), this sequence (m=32), A019814 (m=36).

RealDigits[Sin[Pi/32], 10, 100, -1][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
imag(I^(1/16))
```
PARI
```
sin(Pi/32)
```
PARI
```
sqrt(2-sqrt(2+sqrt(2+sqrt(2))))/2
```

numerical_approx(sin(pi/32), digits=123) # G. C. Greubel, Sep 30 2022

Equals (1/2) * sqrt(2-sqrt(2+sqrt(2+sqrt(2)))).

One of the 16 real roots of -128*x^2 +2688*x^4 -21504*x^6 +84480*x^8 +32768*x^16 -131072*x^14 +212992*x^12 -180224*x^10 +1 =0. - R. J. Mathar, Aug 29 2025

A206769 Decimal expansion of the Fresnel integral Integral_{x=0..oo} sin(x^4) dx.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 10 2013

Imaginary part associated with A206161.

			0.3468652110238094960420351000471...

Wikipedia, Fresnel Integral.

Cf. A068465, A068467, A093954, A182168, A206161.

evalf(Pi*sin(Pi/8)/GAMMA(3/4)/2^(3/2)) ;

RealDigits[Pi * Sin[Pi/8] / (2^(3/2) * Gamma[3/4]), 10, 120][[1]] (* Amiram Eldar, Aug 23 2024 *)

Equals A093954 * A182168 / A068465.

(this constant)^2 + A206161 ^2 = A068467 ^2.

cp's OEIS Frontend

A232736 Decimal expansion of sin(Pi/14), or the imaginary part of (-1)^(1/7).

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A101464 Decimal expansion of sqrt(2-sqrt(2)), edge length of a regular octagon with circumradius 1.

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A154739 Decimal expansion of sqrt(1 - 1/sqrt(2)), the abscissa 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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A188582 Decimal expansion of sqrt(2) - 1.

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A232738 Decimal expansion of the imaginary part of I^(1/8), or sin(Pi/16).

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A280585 Decimal expansion of 8*sin(Pi/8).

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