A019851 -id:A019851 - 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

A019875 Decimal expansion of sine of 66 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

A quartic number with denominator 2. - Charles R Greathouse IV, Aug 27 2017

			0.913545457...

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

Cf. A019821, A019842, A019866, A019887.

RealDigits[N[Cos[2*Pi/15], 110]][[1]] (* Wesley Ivan Hurt, Dec 27 2021 *)

cos(2*Pi/15) \\ Charles R Greathouse IV, Aug 27 2017

Equals cos(2*Pi/15) = 2*A019887^2 - 1 = 1 - 2*A019821^2. - R. J. Mathar, Jun 18 2006
Equals 2*A019842*A019866. - R. J. Mathar, Jan 17 2021
Largest of the 4 real-valued roots of 16*x^4-8*x^3-16*x^2+8*x+1=0. (Other A019851, -A019815, -A019887). - R. J. Mathar, Sep 04 2025

A019815 Decimal expansion of sine of 6 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Decimal expansion of 1/8 (-1 - sqrt(5) + sqrt(6*(5 - sqrt(5)))). - Artur Jasinski, Oct 28 2008

			sin(Pi/30) = 0.10452846...

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

First[RealDigits[1/8 (-1 - Sqrt[5] + Sqrt[6 (5 - Sqrt[5])]), 10, 100]] (* Artur Jasinski, Oct 28 2008 *)
RealDigits[Sin[6 Degree],10,120][[1]] (* Harvey P. Dale, Apr 27 2012 *)

sin(Pi/30) \\ Charles R Greathouse IV, Jan 30 2016

Equals cos(7*Pi/15) = -cos(8*Pi/15) = 2F1(6/5,-1/5;1/2;3/4) / 2 = -2F1(13/10,-3/10;1/2;3/4) / 2. - R. J. Mathar, Oct 27 2008
Equals 2*A019812*A019896. - R. J. Mathar, Jan 17 2021
Smallest positive of the 4 real-valued roots of 16*x^4+8*x^3-16*x^2-8*x+1=0. (Other A019887, -A019875, -A019851) - 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

A306603 a(n) = (2 cos(Pi/15))^n + (2 cos(7 Pi/15))^n + (2 cos(11 Pi/15))^n + (2 cos(13 Pi/15))^n.

Original entry on oeis.org

4, -1, 9, -1, 29, 4, 99, 34, 349, 179, 1254, 824, 4559, 3574, 16704, 15004, 61549, 61709, 227799, 250229, 846254, 1004149, 3153984, 3997399, 11788879, 15812504, 44178624, 62229509, 165946124, 243873904, 624650004, 952400599, 2355748909, 3708579599

Offset: 0

Greg Dresden, Feb 27 2019

a(n) is obtained from the Girard-Waring formula for the sum of powers of N = 4 indeterminates (see A324602), with the elementary symmetric functions e_1 = -1, e_2 = -4, e_3 = -4 and e_4 = 1. The arguments are e_j(x_1, x_2, x_3, x_4), for j = 1..4, with the zeros {x_i}A187360,%20for%20n%20=%2015),%20appearing%20to%20the%20power%20n%20in%20the%20formula%20given%20above.%20-%20_Wolfdieter%20Lang">{i=1..4} of the minimal polynomial of 2*cos(Pi/15) (see A187360, for n = 15), appearing to the power n in the formula given above. - _Wolfdieter Lang, May 08 2019

Index entries for linear recurrences with constant coefficients, signature (-1,4,4,-1).

Cf. A019887 (cos(Pi/15)), A019815 (cos(7*Pi/15)), A019851 (cos(11*Pi/15)), A019875 (cos(13*Pi/15)), A187360, A324602.

Table[Sum[(2.0 Cos[k Pi/15])^n, {k, {1, 7, 11, 13}}] // Round, {n, 1, 30}]
LinearRecurrence[{-1,4,4,-1},{4,-1,9,-1},40] (* Harvey P. Dale, Jun 02 2024 *)

G.f.: (4*x^3+8*x^2-3*x-4)/(-x^4+4*x^3+4*x^2-x-1). - Alois P. Heinz, Feb 27 2019

a(n) = -a(n-1) + 4*a(n-2) + 4*a(n-3) -a(n-4). - Greg Dresden, Feb 27 2019

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

A306610 a(n) = (2cos(Pi/15))^(-n) + (2cos(7Pi/15))^(-n) + (2cos(11Pi/15))^(-n) + (2cos(13*Pi/15))^(-n), for n >= 1.

Original entry on oeis.org

4, 24, 109, 524, 2504, 11979, 57299, 274084, 1311049, 6271254, 29997829, 143491199, 686373809, 3283190949, 15704770004, 75121978804, 359337430474, 1718849676159, 8221921677724, 39328626006254, 188124003629279, 899869747188249, 4304424455586134

Offset: 1

Greg Dresden, Feb 28 2019

-a(n) is the coefficient of x in the minimal polynomial for (2*cos(Pi/15))^n, for n >= 1. The coefficients of -x^3 are A306603(n), and those of x^2 are A306611(n).

a(n) is obtained from the Girard-Waring formula for the sum of powers of N = 4 indeterminates (see A324602), with the elementary symmetric functions e_1 = 4, e_2 = -4, e_3 = -1 and e_4 = 1. The arguments are e_j(1/x_1, 1/x_2, 1/x_3, 1/x_4), for j = 1..4, with the zeros {x_i}{i=1..4} of the minimal polynomial of 2*cos(Pi/15), appearing under the negative powers of the formula given above. - _Wolfdieter Lang, May 08 2019

Index entries for linear recurrences with constant coefficients, signature (4,4,-1,-1).

Cf. A019887 (cos(Pi/15)), A019815 (cos(7*Pi/15)), A019851 (cos(11*Pi/15)), A019875 (cos(13*Pi/15)), A306603 (positive powers of these cosines), A306611, A324602.

Table[Round[N[Sum[(2 Cos[k Pi/15])^(-n), {k,{1,7,11,13}}],50]],{n,1,30}]

a(n) = 4a(n-1) + 4a(n-2) - a(n-3) - a(n-4).
G.f.: x*(-4x^3 -3x^2 +8x +4)/(x^4 +x^3 -4x^2 -4x +1).
a(n) = round((2*cos(7*Pi/15))^(-n)) for n >= 3.

A019940 Decimal expansion of tangent of 42 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the decimal expansion of cotangent of 48 degrees. - Ivan Panchenko, Sep 01 2014

			0.900404044297839945120477203885371702076466211299485282427079...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants

Cf. A019851 (sine of 42 degrees)

SetDefaultRealField(RealField(100)); R:= RealField(); Tan(7*Pi(R)/30); // G. C. Greubel, Nov 25 2018

RealDigits[Tan[42 Degree],10,120][[1]] (* Harvey P. Dale, Sep 05 2012 *)
RealDigits[Tan[7*Pi/30], 10, 100][[1]] (* G. C. Greubel, Nov 25 2018 *)

default(realprecision, 100); tan(7*Pi/30) \\ G. C. Greubel, Nov 25 2018

numerical_approx(tan(7*pi/30), digits=100) # G. C. Greubel, Nov 25 2018

Equals sqrt(7 + 2*sqrt(5) - 2*sqrt(3*(5 + 2*sqrt(5)))). - G. C. Greubel, Nov 25 2018

cp's OEIS Frontend

A010503 Decimal expansion of 1/sqrt(2).

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A019875 Decimal expansion of sine of 66 degrees.

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A019815 Decimal expansion of sine of 6 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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A306603 a(n) = (2 cos(Pi/15))^n + (2 cos(7 Pi/15))^n + (2 cos(11 Pi/15))^n + (2 cos(13 Pi/15))^n.

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A306610 a(n) = (2cos(Pi/15))^(-n) + (2cos(7Pi/15))^(-n) + (2cos(11Pi/15))^(-n) + (2cos(13*Pi/15))^(-n), for n >= 1.