cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 18 results. Next

A010503 Decimal expansion of 1/sqrt(2).

Original entry on oeis.org

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

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Author

Keywords

Comments

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

Examples

			0.7071067811865475...
		

References

  • 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.

Crossrefs

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).

Programs

  • Magma
    1/Sqrt(2); // Vincenzo Librandi, Feb 21 2016
  • Maple
    Digits:=100; evalf(1/sqrt(2)); Wesley Ivan Hurt, Mar 27 2014
  • Mathematica
    N[ 1/Sqrt[2], 200]
    RealDigits[1/Sqrt[2],10,120][[1]] (* Harvey P. Dale, Mar 25 2019 *)
  • PARI
    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
    

Formula

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

Extensions

More terms from Harry J. Smith, Jun 02 2009

A019896 Decimal expansion of sine of 87 degrees.

Original entry on oeis.org

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

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Comments

Equals sin(29*Pi/60). - Wesley Ivan Hurt, Sep 01 2014
An algebraic number of degree 16 and denominator 2. - Charles R Greathouse IV, Aug 27 2017

Examples

			0.998629534754573873784492058439436580590952290767785532441441...
		

Programs

Formula

Equals cos(Pi/60) = [5-sqrt(5)]*[1+sqrt(3)]*[2-sqrt(3)+sqrt{5+2*sqrt(5)}]/[8*sqrt(10)] = sqrt[(1+A019893)/2]. - R. J. Mathar, Jun 18 2006

A019824 Decimal expansion of sine of 15 degrees.

Original entry on oeis.org

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

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Author

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Comments

Also the imaginary part of i^(1/6). - Stanislav Sykora, Apr 25 2012

Examples

			0.258819045102520762348898837624048328349068901319930513814003207315...
		

Crossrefs

Programs

Formula

Equals (sqrt(3)-1)/(2*sqrt(2)) = (A002194 -1) * A020765 = sin(Pi/12). - R. J. Mathar, Jun 18 2006
Equals 2F1(9/8,-1/8;1/2;3/4) / 2 = - 2F1(11/8,-3/8;1/2;3/4) / 2 = cos(5*Pi/12). - R. J. Mathar, Oct 27 2008
Equals sqrt(2 - sqrt(3))/2 = (1/2) * A101263. - Amiram Eldar, Aug 05 2020
This^2 + A019884^2=1. - R. J. Mathar, Aug 31 2025
Smallest positive of the 4 real-valued roots of 16*x^4-16*x^2+1=0. - R. J. Mathar, Aug 31 2025

A019884 Decimal expansion of sine of 75 degrees.

Original entry on oeis.org

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

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Author

Keywords

Comments

Also the real part of i^(1/6). - Stanislav Sykora, Apr 25 2012
Length of one side of the new Type 15 Convex Pentagon. - Michel Marcus, Aug 04 2015

Examples

			0.96592582628906828674974319972889736763390483900840455040234307631042...
		

Crossrefs

Cf. A120683.

Programs

Formula

Equals cos(Pi/12) = [1+sqrt(3)]/[2*sqrt(2)] = A090388 * A020765. - R. J. Mathar, Jun 18 2006
Equals A019859 * A019874 + A019834 * A019849 = A019881 * A019896 + A019812 * A019827 . - R. J. Mathar, Jan 27 2021
Equals 1/(sqrt(6) - sqrt(2)) = 1/A120683. - Amiram Eldar, Aug 04 2022
Largest of the 4 real-valued roots of 16*x^4 -16*x^2 +1=0. - R. J. Mathar, Aug 29 2025
4*this^3 -3*this = A010503. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/8,1/8 ;1/2;3/4) = 2F1(-1/6,1/6;1/2;1/2). - R. J. Mathar, Aug 31 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

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Author

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Comments

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

Examples

			sin(Pi/30) = 0.10452846...
		

Programs

Formula

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

A019818 Decimal expansion of sine of 9 degrees.

Original entry on oeis.org

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

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Author

Keywords

Comments

Also the imaginary part of i^(1/10). - Stanislav Sykora, Apr 25 2012
An algebraic number of degree 8 and denominator 2. - Charles R Greathouse IV, Aug 27 2017

Examples

			sin(Pi/20) = 0.1564344650402308690101053194671668923138...
		

Crossrefs

Programs

Formula

sqrt(8-2*sqrt(10+2*sqrt(5)))/4 = sqrt((1/2)*(1 - sqrt((1/8)*(5 + sqrt(5))))). - Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 13 2006
Equals A019815*A019896 + A019812*A019893. - R. J. Mathar, Jan 27 2021
This^2 + A019890^2=1. - R. J. Mathar, Aug 31 2025
Smallest positive of the 8 real-values roots of 256*x^8 -512*x^6 +304*x^4 -48*x^2+1=0. - R. J. Mathar, Aug 31 2025

A019890 Decimal expansion of sine of 81 degrees.

Original entry on oeis.org

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

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Author

Keywords

Comments

Also the real part of i^(1/10). - Stanislav Sykora, Apr 25 2012
Equals sin(9*Pi/20). - Wesley Ivan Hurt, Sep 01 2014
An algebraic number of degree 8 and denominator 2. - Charles R Greathouse IV, Aug 27 2017

Examples

			0.98768834059513772619004024769343726075840686158988043492390480163...
		

Programs

Formula

Equals cos(Pi/20) = sqrt((1+A019881)/2) = sqrt(1-A019818^2) = sqrt(5-sqrt(5))*(sqrt(5)+sqrt(5+2*sqrt(5)))/(4*sqrt(5)). - R. J. Mathar, Jun 18 2006
Root of 256*x^8 -512*x^6 +304*x^4 -48*x^2+1=0. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/10,1/10;1/2;1/2). - R. J. Mathar, Aug 31 2025

A019836 Decimal expansion of sine of 27 degrees.

Original entry on oeis.org

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

Views

Author

Keywords

Comments

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

Programs

Formula

Equals cos(7*Pi/20). 2*this^2-1 = -A019845. - R. J. Mathar, Aug 29 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

Views

Author

Keywords

Comments

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

Programs

Formula

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

A019878 Decimal expansion of sine of 69 degrees.

Original entry on oeis.org

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

Views

Author

Keywords

Comments

An algebraic number of degree 16 and denominator 2. - Charles R Greathouse IV, Nov 06 2017

Programs

Formula

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