A010503 - OEIS

A010503

Decimal expansion of 1/sqrt(2).

107

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

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OFFSET

0,1

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

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.

LINKS

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.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 545.

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.

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

Conjecture: Equals Sum_{k>=0} A139820(k) / exp(k*Pi). - Simon Plouffe, Sep 09 2025

From Stefano Spezia, May 17 2026: (Start)

Equals Sum_{k>0} (2*k - 1)!!*k/((2*k)!!*2^k).

Equals cos(Pi/12) - sin(Pi/12). (End)

EXAMPLE

0.7071067811865475...

MAPLE

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

MATHEMATICA

N[ 1/Sqrt[2], 200]

(* Alternative: *)

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

PROG

(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

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

CROSSREFS

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, A139820.

Sequence in context: A036479 A085966 A010678 * A392558 A335727 A381979

Adjacent sequences: A010500 A010501 A010502 * A010504 A010505 A010506

KEYWORD

nonn,cons,easy,changed

AUTHOR