%I #161 Oct 16 2024 09:20:59

%S 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,

%T 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,

%U 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

%N Decimal expansion of 1/sqrt(2).

%C The decimal expansion of sqrt(50) = 5*sqrt(2) = 7.0710678118654752440... gives essentially the same sequence.

%C Also real and imaginary part of the square root of the imaginary unit. - _Alonso del Arte_, Jan 07 2011

%C 1/sqrt(2) = (1/2)^(1/2) = (1/4)^(1/4) (see the comments in A072364).

%C 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

%C 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

%C 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

%C 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

%C Radius of midsphere (tangent to edges) in a cube with unit edges. - _Stanislav Sykora_, Mar 27 2014

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Section 1.1, pp. 1-3.

%H Harry J. Smith, <a href="/A010503/b010503.txt">Table of n, a(n) for n = 0..20000</a>

%H P. C. Fishburn and J. A. Reeds, <a href="http://dx.doi.org/10.1137/S0895480191219350">Bell inequalities, Grothendieck's constant and root two</a>, SIAM J. Discrete Math., Vol. 7, No. 1, Feb. 1994, pp. 48-56.

%H Ovidiu Furdui, <a href="https://rgmia.org/pc/2010/problem1-10.pdf">Problem 1</a>, Problem Corner, Research Group in Mathematical Inequalities and Applications, 2010.

%H Michael Penn, <a href="https://www.youtube.com/watch?v=i76d75nDiK4">A surprisingly convergent limit</a>, YouTube video, 2022.

%H Michael Penn, <a href="https://www.youtube.com/watch?v=Xk2LJsZA_MQ">The infinite fraction of your dreams (nightmare?)</a>, YouTube video, 2022.

%H Jonathan Sondow and D. Marques, <a href="http://arxiv.org/abs/1108.6096">Algebraic and transcendental solutions of some exponential equations</a>, Annales Mathematicae et Informaticae 37 (2010) 151-164; arXiv:1108.6096 [math.NT], 2011, see p. 3 in the link.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigitProduct.html">Digit Product</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Platonic solid">Platonic solid</a>.

%H Donald R. Woods, <a href="http://www.jstor.org/stable/2978051">Problem E 2692</a>, Elementary Problems, The American Mathematical Monthly, Vol. 85, No. 1 (1978), p. 48; <a href="http://www.jstor.org/stable/2321105">A Transcendental Function Satisfy a Duplication Formula</a>, by David Robbins, ibid., Vol. 86, No. 5 (1979), pp. 394-395.

%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>.

%F 1/sqrt(2) = cos(Pi/4) = sqrt(2)/2. - _Eric Desbiaux_, Nov 05 2008

%F a(n) = 9 - A268682(n). As constants, this sequence is 1 - A268682. - _Philippe Deléham_, Feb 21 2016

%F From _Amiram Eldar_, Jun 29 2020: (Start)

%F Equals sin(Pi/4) = cos(Pi/4).

%F Equals Integral_{x=0..Pi/4} cos(x) dx. (End)

%F Equals (1/2)*A019884 + A019824 * A010527 = A019851 * A019896 + A019812 * A019857. - _R. J. Mathar_, Jan 27 2021

%F Equals hypergeom([-1/2, -3/4], [5/4], -1). - _Peter Bala_, Mar 02 2022

%F 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

%F Equals Product_{k>=0} ((2*k+1)/(2*k+2))^((-1)^A000120(k)) (Woods, 1978). - _Amiram Eldar_, Feb 04 2024

%F From _Stefano Spezia_, Oct 15 2024: (Start)

%F Equals 1 + Sum_{k>=1} (-1)^k*binomial(2*k,k)/2^(2*k) [Newton].

%F Equal Product_{k>=1} 1 - 1/(4*(2*k - 1)^2). (End)

%e 0.7071067811865475...

%p Digits:=100; evalf(1/sqrt(2)); _Wesley Ivan Hurt_, Mar 27 2014

%t N[ 1/Sqrt[2], 200]

%t RealDigits[1/Sqrt[2],10,120][[1]] (* _Harvey P. Dale_, Mar 25 2019 *)

%o (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

%o (Magma) 1/Sqrt(2); // _Vincenzo Librandi_, Feb 21 2016

%Y Cf. A000120, A040042, A072364, A268682.

%Y Cf. A073084 (infinite tetration limit).

%Y Platonic solids circumradii: A010527 (cube), A019881 (icosahedron), A179296 (dodecahedron), A187110 (tetrahedron).

%Y Platonic solids midradii: A020765 (tetrahedron), A020761 (octahedron), A019863 (icosahedron), A239798 (dodecahedron).

%Y Cf. A000217, A010060, A019812, A019824, A019851, A019857, A019884, A019896.

%K nonn,cons,easy

%O 0,1

%A _N. J. A. Sloane_

%E More terms from _Harry J. Smith_, Jun 02 2009