A154739 - OEIS

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

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OFFSET

0,1

COMMENTS

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

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.

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

EXAMPLE

0.541196100146196984399723205366...

MATHEMATICA

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

PROG

(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

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.

Sequence in context: A132707 A213658 A046575 * A321044 A136564 A136042

Adjacent sequences: A154736 A154737 A154738 * A154740 A154741 A154742

KEYWORD

nonn,cons,easy

AUTHOR

Stuart Clary, Jan 14 2009

EXTENSIONS

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

STATUS

approved