A154739 - OEIS

%I #21 Feb 04 2025 14:52:20

%S 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,

%T 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,

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

%N 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.

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

%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="/A154739/b154739.txt">Table of n, a(n) for n = 0..5000</a>

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

%F From _Amiram Eldar_, Nov 22 2024: (Start)

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

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

%F Equals 1/A179260 = sqrt(A268682). - _Hugo Pfoertner_, Nov 22 2024

%e 0.541196100146196984399723205366...

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

%o (PARI) sqrt(1 - 1/sqrt(2)) \\ _G. C. Greubel_, Sep 23 2017

%o (PARI) polrootsreal(2*x^4-4*x^2+1)[3] \\ _Charles R Greathouse IV_, Feb 04 2025

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

%Y Cf. A154740, A154741 and A154742 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. A002193, A016825, A179260, A182168, A268682.

%K nonn,cons,easy,changed

%O 0,1

%A _Stuart Clary_, Jan 14 2009

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