A144981 -id:A144981 - cp's OEIS frontend

A232737 Decimal expansion of the real part of I^(1/8), or cos(Pi/16).

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

Offset: 0

Stanislav Sykora, Nov 29 2013

The corresponding imaginary part is in A232738.

			0.9807852804032304491261822361342390369739337308933360950029160885453...

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. A232738 (imaginary part), A010503 (real(I^(1/2))), A010527 (real(I^(1/3))), A144981 (real(I^(1/4))), A019881 (real(I^(1/5))), A019884 (real(I^(1/6))), A232735 (real(I^(1/7))), A019889 (real(I^(1/9))), A019890 (real(I^(1/10))).

RealDigits[Cos[Pi/16], 10, 120][[1]] (* Amiram Eldar, Jun 29 2023 *)

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

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

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

Equals (1/2) * sqrt(2+sqrt(2+sqrt(2))). - Seiichi Manyama, Apr 04 2021
Root of 128*x^8 -256*x^6 +160*x^4 -32*x^2 +1 = 0. - R. J. Mathar, Aug 29 2025
2*this^2 -1 = A144981. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/8,1/8;1/2;1/2). - R. J. Mathar, Aug 31 2025

A144982 Decimal expansion of cos(Pi/24) = cos(7.5 degrees).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Sep 28 2008

Octic number of denominator 2 and minimal polynomial 256x^8 - 512x^6 + 320x^4 - 64x^2 + 1. - Charles R Greathouse IV, May 13 2019

			Equals 0.9914448613738104111445575269285628712777382744...

Index entries for algebraic numbers, degree 8

evalf( sqrt(2*sqrt(2)+sqrt(3)+1)/2^(5/4)) ;

RealDigits[ Sqrt[2 + Sqrt[2 + Sqrt[3]]]/2, 10, 105] // First (* Jean-François Alcover, Feb 20 2013 *)
RealDigits[Cos[Pi/24],10,120][[1]] (* Harvey P. Dale, Feb 11 2024 *)

cos(Pi/24) \\ Charles R Greathouse IV, May 13 2019

sqrt(2*sqrt(2)+sqrt(3)+1)/2^(5/4) =sqrt(A010466+A090388)/A011027.
Equals 2F1(9/16,7/16;1/2;3/4) / 2 . - R. J. Mathar, Oct 27 2008
4*this^3 -3*this = A144981. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/16,1/16;1/2;3/4) = 2F1(-1/12,1/12;1/2;1/2). - R. J. Mathar, Aug 31 2025

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

A206161 Decimal expansion of the Fresnel integral int_{x=0..infinity} cos(x^4) dx.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 10 2013

			0.83740669676908648308360272218083226...

Wikipedia, Fresnel Integral

Cf. A204067.

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

RealDigits[ Sqrt[2 + Sqrt[2]]*Gamma[1/4]/8, 10, 72] // First (* Jean-François Alcover, Feb 20 2013 *)

Equals A093954 * A144981 / A068465 .

A343056 Decimal expansion of the real part of i^(1/16), or cos(Pi/32).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

			0.9951847266721968862448369...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Leon D. Fairbanks, Powers of Cosine and Sine, arXiv:2308.04437 [math.GM], 2023. See p. 3.
Wikipedia, Trigonometric constants expressed in real radicals.
Index entries for algebraic numbers, degree 16

cos(Pi/m): A010503 (m=4), A019863 (m=5), A010527 (m=6), A073052 (m=7), A144981 (m=8), A019879 (m=9), A019881 (m=10), A019884 (m=12), A232735 (m=14), A019887 (m=15), A232737 (m=16), A210649 (m=17), A019889 (m=18), A019890 (m=20), A144982 (m=24), A019893 (m=30). this sequence (m=32), A019894 (m=36).

R:= RealField(127); Cos(Pi(R)/32); // G. C. Greubel, Sep 30 2022

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

PARI
```
real(I^(1/16))
```
PARI
```
cos(Pi/32)
```
PARI
```
sqrt(2+sqrt(2+sqrt(2+sqrt(2))))/2
```

numerical_approx(cos(pi/32), digits=122) # G. C. Greubel, Sep 30 2022

Equals (1/2) * sqrt(2+sqrt(2+sqrt(2+sqrt(2)))).
Satisfies 32768*x^16 -131072*x^14 +212992*x^12 -180224*x^10 +84480*x^8 -21504*x^6 +2688*x^4 -128*x^2 +1 = 0. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/16,1/16;1/2;1/2). - R. J. Mathar, Aug 31 2025

A343060 Decimal expansion of tan(Pi/16).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

Smallest positive of the 4 real-valued roots of x^4+4*x^3-6*x^2-4*x+1=0 (Other: tan(5*Pi/16) = 1.49660.., -tan(7*Pi/16) = -5.027339.., -tan(3*Pi/16)= -0.6681786...) - R. J. Mathar, Sep 06 2025

			0.19891236737965800691159762264...

Cf. A232738 (sin(Pi/16)), A232737 (cos(Pi/16)), A343057 (tan(Pi/32)).

RealDigits[Tan[Pi/16], 10, 100][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
tan(Pi/16)
```

sqrt((2-sqrt(2+sqrt(2)))/(2+sqrt(2+sqrt(2))))

PARI
```
sqrt(4+2*sqrt(2))-sqrt(2)-1
```

Equals sqrt( (2-sqrt(2+sqrt(2)))/(2+sqrt(2+sqrt(2))) ).
Equals sqrt(4+2*sqrt(2))-sqrt(2)-1.
Equals A182168/(1+A144981). - R. J. Mathar, Sep 06 2025

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A232737 Decimal expansion of the real part of I^(1/8), or cos(Pi/16).

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A144982 Decimal expansion of cos(Pi/24) = cos(7.5 degrees).

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

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A206161 Decimal expansion of the Fresnel integral int_{x=0..infinity} cos(x^4) dx.

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A343056 Decimal expansion of the real part of i^(1/16), or cos(Pi/32).

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A343060 Decimal expansion of tan(Pi/16).

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