A154739 Decimal expansion of sqrt(1 - 1/sqrt(2)), the abscissa 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.

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

Offset: 0

Author

Stuart Clary, Jan 14 2009

Keywords

• nonn
• cons
• easy

Comments

A root of 2*x^4 - 4*x^2 + 1 = 0.

Examples

			0.541196100146196984399723205366...
		

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for algebraic numbers, degree 4.

Crossrefs

Cf. A154743 for the ordinate and A154747 for the radius vector.

Cf. A154740, A154741 and A154742 for the continued fraction and the numerators and denominators of the convergents.

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

Cf. A002193, A016825, A179260, A182168, A268682.

Programs

Mathematica

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

PARI

sqrt(1 - 1/sqrt(2)) \\ G. C. Greubel, Sep 23 2017

PARI

polrootsreal(2*x^4-4*x^2+1)[3] \\ Charles R Greathouse IV, Feb 04 2025

Formula

From Amiram Eldar, Nov 22 2024: (Start)

Equals sqrt(2) * sin(Pi/8) = A002193 * A182168.

Equals Product_{k>=0} (1 - (-1)^k/(4*k+2)) = Product_{k>=1} (1 + (-1)^k/A016825(k)). (End)

Equals 1/A179260 = sqrt(A268682). - Hugo Pfoertner, Nov 22 2024

Extensions

Offset corrected by R. J. Mathar, Feb 05 2009