A020765 -id:A020765 - 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

A010466 Decimal expansion of square root of 8.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Sqrt(8) = 2*sqrt(2) is the length of the longest (rigid) ladder that can be carried horizontally around a right angled corner in a hallway of unit width. - Lekraj Beedassy, Apr 19 2006
Continued fraction expansion is 2 followed by {1, 4} repeated. - Harry J. Smith, Jun 05 2009
This is the second Lagrange number. - Alonso del Arte, Dec 06 2011
Also 2*sqrt(2) is the ratio of the perimeter of a square to its diameter (diagonal length). - Rick L. Shepherd, Dec 29 2016
Uchiyama shows that every interval (n, n + c*n^(1/4)) contains an integer that is the sum of two squares, where c = 2^(3/2). - Michel Marcus, Jan 03 2018
This is the area of the eighth-smallest triangle with integer side lengths (2, 3, 3), or the seventh-smallest triangle if two smaller triangles with the same area are counted only once (see A331251). - Hugo Pfoertner, Feb 12 2020
Diameter of a sphere whose surface area equals 8*Pi. More generally, the square root of x is also the diameter of a sphere whose surface area equals x*Pi. - Omar E. Pol, Feb 13 2020
Sqrt(8) = area between the curves y = sin(x) and y = cos(x) for Pi/4 < x < 5 Pi/4; this is one of infinitely many congruent convex regions bounded by the two curves. - Clark Kimberling, May 03 2020
Area of the regular 8-gon with circumradius =1. - R. J. Mathar, Aug 24 2023

			2.828427124746190097603377448419396157139343750753896146353359475981464...
Sqrt(8) = sqrt(1+2*i*sqrt(2)) + sqrt(1-2*i*sqrt(2)) = sqrt(1/2+2*i*sqrt(3)) + sqrt(1/2-2*i*sqrt(3)), where i=sqrt(-1). - _Bruno Berselli_, Nov 20 2012
1 + 3/4 + 3*5/(4*8) + 3*5*7/(4*8*12) + 3*5*7*9/(4*8*12*16) + ... - _Bruno Berselli_, Mar 16 2014

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 187.
S. R. Finch, Moving Sofa Constant, Sect. 8.12 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 519-523, 2003.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Jason Kimberley, Index of expansions of sqrt(d) in base b
R. J. Nemiroff & J. Bonnell, The first 1 million digits of the square root of 8
R. J. Nemiroff & J. Bonnell, Plouffe's Inverter, The first 1 million digits of the square root of 8
Ana Rechtman, Juin 2023, 3e défi, Images des Mathématiques, CNRS, 2023.
S. Uchiyama, On the distribution of integers representable as a sum of two h-th powers, J. Fac. Sci. Hokkaido Univ. Ser. I, 18, 124-127, 1964/1965.
Eric Weisstein's World of Mathematics, Moving Ladder Problem
Index entries for algebraic numbers, degree 2

Cf. A040005 (continued fraction).

Cf. A331250, A331251.

SetDefaultRealField(RealField(100)); Sqrt(8); // Vincenzo Librandi, Feb 13 2020

evalf(2^(3/2)) ; # R. J. Mathar, Jul 15 2013

RealDigits[N[Sqrt[8],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)

default(realprecision, 20080); x=sqrt(8); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010466.txt", n, " ", d));  \\ Harry J. Smith, Jun 02 2009

Equals 1 + Sum_{n>=1} ( Product_{k=1..n} (2k+1)/(4k) ). - Bruno Berselli, Mar 16 2014
Equals 2*A002193. - R. J. Mathar, Jan 14 2021
From Peter Bala, Mar 01 2022: (Start)
Equals 3*Sum_{n >= 0} (1/(4*n+1) - 1/(4*n-3))*binomial(1/2,n). Cf. A002580 and A175576.
Equals 4*hypergeom([-1/2, -3/4], [5/4], -1). (End)
Equals 8 * A020765. - R. J. Mathar, Aug 24 2023

A020760 Decimal expansion of 1/sqrt(3).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

If the sides of a triangle form an arithmetic progression in the ratio 1:1+d:1+2d then when d=1/sqrt(3) it uniquely maximizes the area of the triangle. This triangle has approximate internal angles 25.588 degs, 42.941 degs, 111.471 degs. - Frank M Jackson, Jun 15 2011
When a cylinder is completely enclosed by a sphere, it occupies a fraction f of the sphere volume. The value of f has a trivial lower bound of 0, and an upper bound which is this constant. It is achieved iff the cylinder diameter is sqrt(2) times its height, and the sphere is circumscribed to it. A similar constant can be associated with any n-dimensional geometric shape. For 3D cuboids it is A165952. - Stanislav Sykora, Mar 07 2016
The ratio between the thickness and diameter of a dynamically fair coin having an equal probability, 1/3, of landing on each of its two faces and on its side after being tossed in the air. The calculation is based on the dynamic of rigid body (Yong and Mahadevan, 2011). See A020765 for a simplified geometrical solution. - Amiram Eldar, Sep 01 2020
The coefficient of variation (relative standard deviation) of natural numbers: Limit_{n->oo} sqrt((n-1)/(3*n+3)) = 1/sqrt(3). - Michal Paulovic, Mar 21 2023

			0.577350269189625764509148780501957455647601751270126876018602326....

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 8.4.3 and 8.17, pp. 495, 531.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Ee Hou Yong and L. Mahadevan, Probability, geometry, and dynamics in the toss of a thick coin, American Journal of Physics, Vol. 79, No. 12 (2011), pp. 1195-1201, preprint, arXiv:1008.4559 [physics.class-ph], 2010-2011.
Index entries for algebraic numbers, degree 2.

Cf. A002194 (sqrt(3)), A010701 (1/3).

Cf. A002193, A047235, A165952, A195696, A040001.

evalf(1/sqrt(3)); # Michal Paulovic, Mar 21 2023

RealDigits[N[1/Sqrt[3],200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

\\ Works in v2.15.0; n = 100 decimal places
my(n=100); digits(floor(10^n/quadgen(12))) \\ Michal Paulovic, Mar 21 2023

Equals 1/A002194. - Michel Marcus, Oct 12 2014
Equals cosine of the magic angle: cos(A195696). - Stanislav Sykora, Mar 07 2016
Equals square root of A010701. - Michel Marcus, Mar 07 2016
Equals 1 + Sum_{k>=0} -(4*k+1)^(-1/2) + (4*k+3)^(-1/2) + (4*k+5)^(-1/2) - (4*k+7)^(-1/2). - Gerry Martens, Nov 22 2022
Equals (1/2)*(2 - zeta(1/2,1/4) + zeta(1/2,3/4) + zeta(1/2,5/4) - zeta(1/2,7/4)). - Gerry Martens, Nov 22 2022
Has periodic continued fraction expansion [0, 1; 1, 2] (A040001). - Michal Paulovic, Mar 21 2023
Equals Product_{k>=1} (1 + (-1)^k/A047235(k)). - Amiram Eldar, Nov 22 2024
Equals tan(Pi/6) = (1/2)/A010527. - R. J. Mathar, Aug 31 2025

A019863 Decimal expansion of sin(3*Pi/10) (sine of 54 degrees, or cosine of 36 degrees).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Midsphere radius of regular icosahedron with unit edges.
Also half of the golden ratio (A001622). - Stanislav Sykora, Jan 30 2014
Andris Ambainis (see Aaronson link) observes that combining the results of Barak-Hardt-Haviv-Rao with Dinur-Steurer yields the maximal probability of winning n parallel repetitions of a classical CHSH game (see A201488) asymptotic to this constant to the power of n, an improvement on the naive probability of (3/4)^n. (All the random bits are received upfront but the players cannot communicate or share an entangled state.) - Charles R Greathouse IV, May 15 2014
This is the height h of the isosceles triangle in a regular pentagon, in length units of the circumscribing radius, formed by a side as base and two adjacent radii. h = sin(3*Pi/10) = cos(Pi/5) (radius 1 unit). - Wolfdieter Lang, Jan 08 2018
Also the limiting value(L) of "r" which is abscissa of the vertex of the parabola  F(n)*x^2 - F(n+1)*x + F(n + 2)(where F(n)=A000045(n) are the Fibonacci numbers and n>0). - Burak Muslu, Feb 24 2021

			0.80901699437494742410229341718281905886015458990288143106772431135263...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Scott Aaronson, The NEW Ten Most Annoying Questions in Quantum Computing (2014).
Boaz Barak, Moritz Hardt, Ishay Haviv, and Anup Rao, Rounding Parallel Repetitions of Unique Games (2008).
Irit Dinur and David Steurer, Analytical approach to parallel repetition, arXiv:1305.1979 [cs.CC], 2013-2014.
Michael Penn, a golden value of cosine., YouTube video, 2021.
Wikipedia, Exact trigonometric constants.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 2.

Cf. A001611, A001622, A019827, A134972, A134944, A134946, A187426, A296182.

Platonic solids midradii: A020765 (tetrahedron), A020761 (octahedron), A010503 (cube), A239798 (dodecahedron).

convert(sin(3*Pi/10),radical); # W. Edwin Clark, May 24 2023
Digits:=100; evalf((1+sqrt(5))/4); # Wesley Ivan Hurt, Mar 27 2014

RealDigits[(1 + Sqrt[5])/4, 10, 111] (* Robert G. Wilson v *)
RealDigits[Sin[54 Degree],10,120][[1]] (* Harvey P. Dale, Apr 21 2018 *)

(1+sqrt(5))/4 \\ Charles R Greathouse IV, Jan 16 2012

Equals (1+sqrt(5))/4 = cos(Pi/5) = sin(3*Pi/10). - R. J. Mathar, Jun 18 2006
Equals 2F1(4/5,1/5;1/2;3/4) / 2 = A019827 + 1/2. - R. J. Mathar, Oct 27 2008
Equals A001622 / 2. - Stanislav Sykora, Jan 30 2014
phi / 2 = (i^(2/5) + i^(-2/5)) / 2 = i^(2/5) - (sin(Pi/5))*i = i^(-2/5) + (sin(Pi/5))*i = i^(2/5) - (cos(3*Pi/10))*i = i^(-2/5) + (cos(3*Pi/10))*i. - Jaroslav Krizek, Feb 03 2014
Equals 1/A134972. - R. J. Mathar, Jan 17 2021
Equals 2*A019836*A019872. - R. J. Mathar, Jan 17 2021
Equals (A094214 + 1)/2 or 1/(2*A094214). - Burak Muslu, Feb 24 2021
Equals hypergeom([-2/5, -3/5], [6/5], -1) = hypergeom([-1/5, 3/5], [6/5], 1) = hypergeom([1/5, -3/5], [4/5], 1). - Peter Bala, Mar 04 2022
Equals Product_{k>=1} (1 - (-1)^k/A001611(k)). - Amiram Eldar, Nov 28 2024
Equals 2*A134944 = 3*A134946 = A187426-11/10 = A296182-1. - Hugo Pfoertner, Nov 28 2024
Equals A134945/4. Root of 4*x^2-2*x-1=0. - R. J. Mathar, Aug 29 2025

A201488 Decimal expansion of maximal success probability of the CHSH game.

Original entry on oeis.org

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

Offset: 0

Charles R Greathouse IV, Dec 02 2011

A referee chooses two random bits and gives one to each of two players who share an entangled quantum state but are not permitted to communicate. The players each choose a bit to send to the referee. If both of the bits from the referee are 1, then the players win if their chosen bits are different; otherwise they win if their chosen bits are the same. The best classical win probability is 3/4, but this can be improved in a quantum setting.
The optimality of this probability follows from Tsirelson's inequality and is implicit in the CHSH paper.
Ratio of leg length to base length in an isosceles triangle with the property that the areas of the two smaller excircles sum up to the area of the third excircle. - Martin Janecke, Aug 05 2012
Weight factor in Laguerre-Gauss quadrature, corresponding to the smallest root of the Laguerre polynomial of degree 2. - A.H.M. Smeets, May 15 2025

			0.853553390593273762200422181052424519642417968844237...

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed Experiment to Test Local Hidden-Variable Theories, Phys. Rev. Lett. 23 (1969), pp. 880-884.
Claude Crépeau, Louis Salvail, Jean-Raymond Simard, and Alain Tapp, Classical and quantum strategies for two-prover bit commitments, Ninth Workshop on Quantum Information Processing (2006).
Erica Klarreich, The Proof in the Quantum Pudding, Quanta Magazine, 21 Aug 2013.
Stephen J. Summers and Reinhard Werner, The vacuum violates Bell's inequalities, Physics Letters A 110:5 (1985), pp. 257-259.
Wikipedia, CHSH inequality
Wikipedia, Quantum pseudo-telepathy
Wikipedia, Tsirelson's bound
Index entries for algebraic numbers, degree 2.

Cf. A020765, A019863.

RealDigits[Cos[Pi/8]^2,10,120][[1]] (* Harvey P. Dale, Jan 21 2012 *)

cos(Pi/8)^2 \\ Charles R Greathouse IV, Dec 02 2011

Equals cos^2(Pi/8) = (1 + 1/sqrt(2))/2.

Equals (theta_3(0, q^2)/theta_3(0, q))^2 where q = 1/e^Pi. - Michael Somos, Dec 02 2022

A020761 Decimal expansion of 1/2.

Original entry on oeis.org

5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Real part of all nontrivial zeros of the Riemann zeta function (assuming the Riemann hypothesis to be true). - Alonso del Arte, Jul 02 2011
Radius of a sphere with surface area Pi. - Omar E. Pol, Aug 09 2012
Radius of the midsphere (tangent to the edges) in a regular octahedron with unit edges. Also radius of the inscribed sphere (tangent to faces) in a cube with unit edges. - Stanislav Sykora, Mar 27 2014
Construct a rectangle of maximal area inside an arbitrary triangle. The ratio of the rectangle's area to the triangle's area is 1/2. - Rick L. Shepherd, Jul 30 2014

			1/2 = 0.50000000000000...

Michael Penn, A creative approach to a scary looking integral., YouTube video, 2020.
Michael Penn, I really like this sum!, YouTube video, 2021.
Wikipedia, Riemann zeta function
Wikipedia, Platonic solid
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. In platonic solids:
midsphere radii:
A020765 (tetrahedron),
A010503 (cube),
A019863 (icosahedron),
A239798 (dodecahedron);
insphere radii:
A020781 (tetrahedron),
A020763 (octahedron),
A179294 (icosahedron),
A237603 (dodecahedron).

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

RealDigits[1/2, 10, 128][[1]] (* Alonso del Arte, Dec 13 2013 *)
LinearRecurrence[{1},{5,0},99] (* Ray Chandler, Jul 15 2015 *)

{ default(realprecision); x=1/2*10; for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", ")) } \\ Felix Fröhlich, Jul 24 2014

a(n) = 5*(n==0); \\ Michel Marcus, Jul 25 2014

Equals Sum_{k>=1} (1/3^k). Hence 1/2 = 0.1111111111111... in base 3.
Cosine of 60 degrees, i.e., cos(Pi/3).
-zeta(0), zeta being the Riemann function. - Stanislav Sykora, Mar 27 2014
a(0) = 5; a(n) = 0, n > 0. - Wesley Ivan Hurt, Mar 27 2014
a(n) = 5 * floor(1/(n + 1)). - Wesley Ivan Hurt, Mar 27 2014
Equals 2*A019824*A019884. - R. J. Mathar, Jan 17 2021

A020829 Decimal expansion of 1/sqrt(72) = 1/(3*2^(3/2)) = sqrt(2)/12.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Volume of regular tetrahedron with unit edge. - Stanislav Sykora, May 31 2012
In the dragon curve fractal, (5/6)*sqrt(2) = 1.1785.... is the maximum distance of any point from curve start.  Such a maximum must be to a vertex of the convex hull.  Hull vertices are shown by Benedek and Panzone (theorem 3, page 85) and their P8 = 7/6 - (1/6)i at distance sqrt((7/6)^2 + (1/6)^2) is the maximum. - Kevin Ryde, Nov 22 2019
With offset 1, volume of a triangular cupola (Johnson solid J_3) with unit edges. - Paolo Xausa, Aug 04 2025

			0.117851130197757920733474...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Agnes I. Benedek and Rafael Panzone, On Some Notable Plane Sets, II: Dragons, Revista de la Unión Matemática Argentina, volume 39, numbers 1-2, 1994, pages 76-90.
Wikipedia, Platonic solid.
Wikipedia, Tetrahedron.
Wikipedia, Triangular cupola.
Index entries for algebraic numbers, degree 2

Cf. A131594 (regular octahedron volume), A102208 (regular icosahedron volume), A102769 (regular dodecahedron volume).

Cf. A010524, A020765, A020775, A378207.

RealDigits[Sqrt[2]/12, 10, 50][[1]] (* G. C. Greubel, Jul 06 2017 *)

sqrt(2)/12 \\ G. C. Greubel, Jul 06 2017

Equals Integral_{x=0..Pi/4} sin(x)^2 * cos(x) dx. - Amiram Eldar, May 31 2021

Equals 1/A010524 = A020765/3 = A020775/2 = A378207/5. - Hugo Pfoertner, Jan 26 2025

A239798 Decimal expansion of the midsphere radius in a regular dodecahedron with unit edges.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Mar 27 2014

In a regular polyhedron, the midsphere is tangent to all edges.

Apart from leading digits the same as A019863 and A019827. - R. J. Mathar, Mar 30 2014

			1.30901699437494742410229341718281905886015458990288143106772431135263...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 2.

Cf. A000045, A001622, A001654, A019827, A019863, A066983, A094884, A104457, A187799.

Midsphere radii in Platonic solids: A020765 (tetrahedron), A020761 (octahedron), A010503 (cube), A019863 (icosahedron).

Digits:=100: evalf((3+sqrt(5))/4); # Wesley Ivan Hurt, Mar 27 2014

RealDigits[GoldenRatio^2/2,10,105][[1]] (* Vaclav Kotesovec, Mar 27 2014 *)

PARI
```
(3+sqrt(5))/4
```

Equals phi^2/2, phi being the golden ratio (A001622).
Equals (3+sqrt(5))/4.
Equals lim_{n->oo} A000045(n)/A066983(n). - Dimitri Papadopoulos, Nov 23 2023
Equals Product_{k>=2} (1 + (-1)^k/A001654(k)). - Amiram Eldar, Dec 02 2024
Equals A094884^2 = A104457/2 = 10/A187799. - Hugo Pfoertner, Dec 02 2024

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

N. J. A. Sloane

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

			0.258819045102520762348898837624048328349068901319930513814003207315...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Mark B. Villarino, Legendre's Singular Modulus, arXiv:2002.07250 [math.HO], 2020.
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 4

Cf. A002194, A020765, A101263.

RealDigits[ Sin[ Pi/12], 10, 111][[1]] (* Or *) RealDigits[(Sqrt[3] - 1)/(2 Sqrt[2]), 10, 111][[1]] (* Robert G. Wilson v *)
RealDigits[Sin[15 Degree],10,120][[1]] (* Harvey P. Dale, Jul 16 2016 *)

sin(Pi/12) \\ Charles R Greathouse IV, Apr 25 2012

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
Equals A019819 * A019894 + A019814 * A019889 = A019821 * A019896 + A019812 * A019887. - R. J. Mathar, Jan 27 2021
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

N. J. A. Sloane

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

			0.96592582628906828674974319972889736763390483900840455040234307631042...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Casey Mann et al, Type 15 Convex Pentagon, Jul 29 2015.
Wikipedia, Trigonometric constants expressed in real radicals.
Index entries for algebraic numbers, degree 4

Cf. A120683.

RealDigits[Sin[75 Degree], 10, 100][[1]] (* Vincenzo Librandi, Aug 11 2014 *)

cos(Pi/12) \\ Charles R Greathouse IV, Apr 25 2012

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

cp's OEIS Frontend

A010503 Decimal expansion of 1/sqrt(2).

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A010466 Decimal expansion of square root of 8.

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A020760 Decimal expansion of 1/sqrt(3).

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A019863 Decimal expansion of sin(3*Pi/10) (sine of 54 degrees, or cosine of 36 degrees).

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A201488 Decimal expansion of maximal success probability of the CHSH game.

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A020761 Decimal expansion of 1/2.

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