A019857 -id:A019857 - cp's OEIS frontend

A010503 Decimal expansion of 1/sqrt(2).

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

Offset: 0

N. J. A. Sloane

The decimal expansion of sqrt(50) = 5*sqrt(2) = 7.0710678118654752440... gives essentially the same sequence.
Also real and imaginary part of the square root of the imaginary unit. - Alonso del Arte, Jan 07 2011
1/sqrt(2) = (1/2)^(1/2) = (1/4)^(1/4) (see the comments in A072364).
If a triangle has sides whose lengths form a harmonic progression in the ratio 1 : 1/(1 + d) : 1/(1 + 2d) then the triangle inequality condition requires that d be in the range -1 + 1/sqrt(2) < d < 1/sqrt(2). - Frank M Jackson, Oct 11 2011
Let s_2(n) be the sum of the base-2 digits of n and epsilon(n) = (-1)^s_2(n), the Thue-Morse sequence A010060, then Product_{n >= 0} ((2*n + 1)/(2*n + 2))^epsilon(n) = 1/sqrt(2). - Jonathan Vos Post, Jun 03 2012
The square root of 1/2 and thus it follows from the Pythagorean theorem that it is the sine of 45 degrees (and the cosine of 45 degrees). - Alonso del Arte, Sep 24 2012
Circumscribed sphere radius for a regular octahedron with unit edges. In electrical engineering, ratio of effective amplitude to peak amplitude of an alternating current/voltage. - Stanislav Sykora, Feb 10 2014
Radius of midsphere (tangent to edges) in a cube with unit edges. - Stanislav Sykora, Mar 27 2014
Positive zero of the Hermite polynomial of degree 2. - A.H.M. Smeets, Jun 02 2025

			0.7071067811865475...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Sections 1.1, 7.5.2, and 8.2, pp. 1-3, 468, 484, 487.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Harry J. Smith, Table of n, a(n) for n = 0..20000
P. C. Fishburn and J. A. Reeds, Bell inequalities, Grothendieck's constant and root two, SIAM J. Discrete Math., Vol. 7, No. 1, Feb. 1994, pp. 48-56.
Ovidiu Furdui, Problem 1, Problem Corner, Research Group in Mathematical Inequalities and Applications, 2010.
Michael Penn, A surprisingly convergent limit, YouTube video, 2022.
Michael Penn, The infinite fraction of your dreams (nightmare?), YouTube video, 2022.
Jonathan Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164; arXiv:1108.6096 [math.NT], 2011, see p. 3 in the link.
Eric Weisstein's World of Mathematics, Digit Product.
Wikipedia, Platonic solid.
Donald R. Woods, Problem E 2692, Elementary Problems, The American Mathematical Monthly, Vol. 85, No. 1 (1978), p. 48; A Transcendental Function Satisfy a Duplication Formula, by David Robbins, ibid., Vol. 86, No. 5 (1979), pp. 394-395.
Index entries for algebraic numbers, degree 2.

Cf. A000120, A040042, A072364, A268682.
Cf. A073084 (infinite tetration limit).
Platonic solids circumradii: A010527 (cube), A019881 (icosahedron), A179296 (dodecahedron), A187110 (tetrahedron).
Platonic solids midradii: A020765 (tetrahedron), A020761 (octahedron), A019863 (icosahedron), A239798 (dodecahedron).
Cf. A000217, A010060, A019812, A019824, A019851, A019857, A019884, A019896.

1/Sqrt(2); // Vincenzo Librandi, Feb 21 2016

Digits:=100; evalf(1/sqrt(2)); Wesley Ivan Hurt, Mar 27 2014

N[ 1/Sqrt[2], 200]
RealDigits[1/Sqrt[2],10,120][[1]] (* Harvey P. Dale, Mar 25 2019 *)

default(realprecision, 20080); x=10*(1/sqrt(2)); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b010503.txt", n, " ", d)); \\ Harry J. Smith, Jun 02 2009

1/sqrt(2) = cos(Pi/4) = sqrt(2)/2. - Eric Desbiaux, Nov 05 2008
a(n) = 9 - A268682(n). As constants, this sequence is 1 - A268682. - Philippe Deléham, Feb 21 2016
From Amiram Eldar, Jun 29 2020: (Start)
Equals sin(Pi/4) = cos(Pi/4).
Equals Integral_{x=0..Pi/4} cos(x) dx. (End)
Equals (1/2)*A019884 + A019824 * A010527 = A019851 * A019896 + A019812 * A019857. - R. J. Mathar, Jan 27 2021
Equals hypergeom([-1/2, -3/4], [5/4], -1). - Peter Bala, Mar 02 2022
Limit_{n->oo} (sqrt(T(n+1)) - sqrt(T(n))) = 1/sqrt(2), where T(n) = n(n+1)/2 = A000217(n) is the triangular numbers. - Jules Beauchamp, Sep 18 2022
Equals Product_{k>=0} ((2*k+1)/(2*k+2))^((-1)^A000120(k)) (Woods, 1978). - Amiram Eldar, Feb 04 2024
From Stefano Spezia, Oct 15 2024: (Start)
Equals 1 + Sum_{k>=1} (-1)^k*binomial(2*k,k)/2^(2*k) [Newton].
Equal Product_{k>=1} 1 - 1/(4*(2*k - 1)^2). (End)
Equals Product_{k>=0} (1 - (-1)^k/(6*k+3)). - Amiram Eldar, Nov 22 2024

More terms from Harry J. Smith, Jun 02 2009

A019851 Decimal expansion of sine of 42 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

This sequence is also decimal expansion of cosine of 48 degrees. - Mohammad K. Azarian, Jun 29 2013

A quartic number with denominator 2. - Charles R Greathouse IV, Nov 05 2017

			0.6691306...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 4

Cf. A019857, A019887, A019830, A019878.

sin(7/30*Pi) \\ Charles R Greathouse IV, Nov 05 2017

Equals cos(4*Pi/15) = (sqrt(5)-1)*(sqrt(3)*sqrt(5+2*sqrt(5))-1)/8 = 8*A019887^4 -8*A019887^2 + 1 = sqrt(1-A019857^2). - R. J. Mathar, Jun 18 2006
Equals 2*A019830*A019878. - R. J. Mathar, Jan 17 2021
A root of 16*x^4-8*x^3-16*x^2+8*x+1 =0. - R. J. Mathar, Aug 31 2025

A019821 Decimal expansion of sine of 12 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 8 and denominator 2. - Charles R Greathouse IV, Aug 27 2017

			0.20791169...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 8

Cf. A019815, A019887, A019893.

RealDigits[N[Sin[Pi/15], 110]][[1]] (* Wesley Ivan Hurt, Dec 27 2021 *)

sin(Pi/15) \\ Charles R Greathouse IV, Aug 27 2017

Equals sin(Pi/15) = sqrt(1-A019887^2) = (sqrt(5)-1)*(sqrt(5+2*sqrt(5)) - sqrt(3))/8. - R. J. Mathar, Jun 18 2006
Equals 2*A019815*A019893. - R. J. Mathar, Jan 17 2021
Smallest positive of the 8 real-valued roots of 256*x^8-448*x^6+224*x^4-32*x^2+1 =0. (Other A019893, A019833, A019857)- R. J. Mathar, Aug 31 2025

A019893 Decimal expansion of sine of 84 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Equals sin(7*Pi/15). - Wesley Ivan Hurt, Sep 01 2014

An algebraic number of degree 8 and denominator 2. - Charles R Greathouse IV, Aug 27 2017

			0.9945218953682733369226919449805703815207920887093194273665588...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 8

Digits:=100: evalf(sin(7*Pi/15)); # Wesley Ivan Hurt, Sep 01 2014

RealDigits[ Cos[Pi/30], 10, 111] (* Robert G. Wilson v *)
RealDigits[Sin[84 Degree],10,120][[1]] (* Harvey P. Dale, May 14 2022 *)

cos(Pi/30) \\ Charles R Greathouse IV, Aug 27 2017

Equals cos(Pi/30) = 2F1(11/20,9/20;1/2;3/4) / 2. - R. J. Mathar, Oct 27 2008
Equals 2*A019851*A019857. - R. J. Mathar, Jan 17 2021
Root of 256*x^8 -448*x^6 +224*x^4 -32*x^2 +1 = 0. - R. J. Mathar, Aug 29 2025
4*this^3 -3*this = A019881. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/20,1/20;1/2;3/4). - R. J. Mathar, Aug 31 2025

A019872 Decimal expansion of sine of 63 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 8 and denominator 2. - Charles R Greathouse IV, Nov 06 2017

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 8

RealDigits[Sin[63 Degree],10,120][[1]] (* Harvey P. Dale, Oct 31 2018 *)

sin(31/60*Pi) \\ Charles R Greathouse IV, Nov 06 2017

Equals A019851 * A019878 + A019830 * A019857 = A010527 * A019896 + A019812 * (1/2). - R. J. Mathar, Jan 27 2021
This^2 + A019836^2=1. - R. J. Mathar, Aug 31 2025
One of the 8 real-valued roots of 256*x^8-512*x^6+304*x^4-48*x^2+1=0. (Other A019890, A019836, A019818) - R. J. Mathar, Aug 31 2025

A019860 Decimal expansion of sine of 51 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 16 and denominator 2. - Charles R Greathouse IV, Nov 05 2017

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 16

RealDigits[Sin[51 Degree],10,120][[1]] (* Harvey P. Dale, Apr 10 2014 *)

sin(17/60*Pi) \\ Charles R Greathouse IV, Nov 05 2017

Equals A019843 * A019882 + A019826 * A019865 = A019857 * A019896 + A019812 * A019851. - R. J. Mathar, Jan 27 2021

A019858 Decimal expansion of sine of 49 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

An algebraic number of degree 48 and denominator 2. - Charles R Greathouse IV, Nov 05 2017

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 48

sin(49/180*Pi) \\ Charles R Greathouse IV, Nov 05 2017

Equals A019851 * A019892 + A019816 * A019857. - R. J. Mathar, Jan 27 2021

A019946 Decimal expansion of tangent of 48 degrees.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Also the decimal expansion of cotangent of 42 degrees. - Ivan Panchenko, Sep 01 2014

			1.11061251482919287014348196416513553257695951039085904818440222...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Wikipedia, Exact trigonometric constants

Cf. A019857 (sine of 48 degrees).

SetDefaultRealField(RealField(100)); R:= RealField(); Tan(4*Pi(R)/15); // G. C. Greubel, Nov 24 2018

RealDigits[Tan[48 Degree],10,120][[1]] (* Harvey P. Dale, Nov 26 2011 *)
RealDigits[Tan[4*Pi/15], 10, 100][[1]] (* G. C. Greubel, Nov 24 2018 *)

default(realprecision, 100); tan(4*Pi/15) \\ G. C. Greubel, Nov 24 2018

numerical_approx(tan(4*pi/15), digits=100) # G. C. Greubel, Nov 24 2018

Equals cot(7*Pi/30) = sqrt(23 - 10*sqrt(5) + 2*sqrt(3*(85 -38*sqrt(5)))). - G. C. Greubel, Nov 24 2018

Let r(n) = (n - 1)/(n + 1) if n mod 4 = 1, (n + 1)/(n - 1) otherwise; then this constant equals with Product_{n>=0} r(30*n+15) = (8/7) * (22/23) * (38/37) * (52/53) ... - Dimitris Valianatos, Sep 14 2019

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A010503 Decimal expansion of 1/sqrt(2).

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A019851 Decimal expansion of sine of 42 degrees.

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A019821 Decimal expansion of sine of 12 degrees.

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A019893 Decimal expansion of sine of 84 degrees.

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A019872 Decimal expansion of sine of 63 degrees.

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A019860 Decimal expansion of sine of 51 degrees.

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A019858 Decimal expansion of sine of 49 degrees.

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