A154747 - OEIS

%I #63 Jul 01 2024 13:16:01

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

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

%U 4,8,5,0,3,2,1,8,3,6,5,1,9,5,4,5,1,3,4,2,4,6,5,9,5

%N Decimal expansion of sqrt(sqrt(2) - 1), the radius vector of the point of bisection of the arc of the unit lemniscate (x^2 + y^2)^2 = x^2 - y^2 in the first quadrant.

%C A root of r^4 + 2 r^2 - 1 = 0.

%C Also real part of sqrt(1 + i)^3, where i is the imaginary unit such that i^2 = -1. - _Alonso del Arte_, Sep 09 2019

%C From _Bernard Schott_, Dec 19 2020: (Start)

%C Length of the shortest line segment which divides a right isosceles triangle with AB = AC = 1 into two parts of equal area; this is the answer to the 2nd problem proposed during the final round of the 18th British Mathematical Olympiad in 1993 (see link BM0 and Gardiner reference).

%C The length of this shortest line segment IJ with I on a short side and J on the hypotenuse is sqrt(sqrt(2)-1), and AI = AJ = 1/sqrt(sqrt(2)) = A228497 (see link Figure for B.M.O. 1993, Problem 2). (End)

%C This algebraic number and its negation equal the real roots of the quartic x^4 + 2*x^2 - 1 (minimal polynomial). The imaginary roots are +A278928*i and -A278928*i. - _Wolfdieter Lang_, Sep 23 2022

%D A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Problem 2, pages 56 and 104-105 (1993).

%D C. L. Siegel, Topics in Complex Function Theory, Volume I: Elliptic Functions and Uniformization Theory, Wiley-Interscience, 1969, page 5

%H G. C. Greubel, <a href="/A154747/b154747.txt">Table of n, a(n) for n = 0..5000</a>.

%H British Mathematical Olympiad 1993, <a href="https://bmos.ukmt.org.uk/home/bmolot.pdf">Problem 2</a>.

%H Bernard Schott, <a href="/A154747/a154747_1.pdf">Figure for B.M.O. 1993, Problem 2</a>.

%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>

%H <a href="/index/Cu#curves">Index to sequences related to curves</a>.

%H <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.

%F From _Peter Bala_, Jul 01 2024: (Start)

%F This constant occurs in the evaluation of Integral_{x = 0..Pi/2} sin^2(x)/(1 + sin^4(x)) dx = Pi/4 * sqrt(sqrt(2) - 1).

%F Equals (1/2)*Sum_{n >= 0} (-1/16)^n * binomial(4*n+2, 2*n+1) (a slowly converging series). (End)

%F Equals 2^(3/4)*sin(Pi/8). - _Vaclav Kotesovec_, Jul 01 2024

%e 0.643594252905582624735443437418...

%t nmax = 1000; First[ RealDigits[ Sqrt[Sqrt[2] - 1], 10, nmax] ]

%o (PARI) sqrt(sqrt(2) - 1) \\ _Michel Marcus_, Dec 10 2016

%Y Cf. A154739 for the abscissa and A154743 for the ordinate.

%Y Cf. A154748, A154749 and A154750 for the continued fraction and the numerators and denominators of the convergents.

%Y Cf. A085565 for 1.311028777..., the first-quadrant arc length of the unit lemniscate.

%Y Cf. A309948 and A309949 for real and imaginary parts of sqrt(1 + i).

%Y Cf. A278928.

%K nonn,cons,easy

%O 0,1

%A _Stuart Clary_, Jan 14 2009

%E Offset corrected by _R. J. Mathar_, Feb 05 2009