A309948 -id:A309948 - cp's OEIS frontend

A154747 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.

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, 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, 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

Offset: 0

Stuart Clary, Jan 14 2009

A root of r^4 + 2 r^2 - 1 = 0.
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
From Bernard Schott, Dec 19 2020: (Start)
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).
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)
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

			0.643594252905582624735443437418...

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

G. C. Greubel, Table of n, a(n) for n = 0..5000.
British Mathematical Olympiad 1993, Problem 2.
Bernard Schott, Figure for B.M.O. 1993, Problem 2.
Index entries for algebraic numbers, degree 4
Index to sequences related to curves.
Index to sequences related to Olympiads.

Cf. A154739 for the abscissa and A154743 for the ordinate.
Cf. A154748, A154749 and A154750 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. A309948 and A309949 for real and imaginary parts of sqrt(1 + i).
Cf. A278928.

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

sqrt(sqrt(2) - 1) \\ Michel Marcus, Dec 10 2016

From Peter Bala, Jul 01 2024: (Start)
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).
Equals (1/2)*Sum_{n >= 0} (-1/16)^n * binomial(4*n+2, 2*n+1) (a slowly converging series). (End)
Equals 2^(3/4)*sin(Pi/8). - Vaclav Kotesovec, Jul 01 2024

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

A309949 Decimal expansion of the imaginary part of the square root of 1 + i.

Original entry on oeis.org

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

Offset: 0

Alonso del Arte, Aug 24 2019

i is the imaginary unit such that i^2 = -1.

Multiplied by -1, this is the imaginary part of the square root of 1 - i. And also the real part of -sqrt(1 + i) - i + sqrt(1 + i)^3, which is a unit in Q(sqrt(1 + i)).

			Im(sqrt(1 + i)) = 0.45508986056222734130435775782247...

Cf. A000108, A010767, A182168, A309948 (real part).

Digits := 120: Re(sqrt(-1 - I))*10^95:
ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Sep 20 2019

RealDigits[Sqrt[1/Sqrt[2] - 1/2], 10, 100][[1]]

imag(sqrt(1+I)) \\ Michel Marcus, Sep 16 2019

Equals sqrt(1/sqrt(2) - 1/2) = 2^(1/4) * sin(Pi/8).
Equals sqrt((sqrt(2) - 1)/2) = A010767 * A182168. - Bernard Schott, Sep 16 2019
Equals Re(sqrt(-1 - i)). - Peter Luschny, Sep 20 2019
Equals Product_{k>=0} ((8*k - 1)*(8*k + 4))/((8*k - 2)*(8*k + 5)). - Antonio Graciá Llorente, Feb 24 2024

A278928 Decimal expansion of sqrt(sqrt(2) + 1).

Original entry on oeis.org

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

Offset: 1

Bobby Jacobs, Dec 01 2016

A quartic integer with minimal polynomial x^4 - 2*x^2 - 1. - Charles R Greathouse IV, Dec 01 2016
Suppose f(n) has the recurrence f(2*n) = f(2*n - 1) + f(2*n - 2) and f(2*n + 1) = f(2*n) + f(2*n - 2), where f(0) and f(1) are not both 0. Then, lim_{n -> oo} f(n)^(1/n) is this constant.
Apart from the first digit, the same as A190283. - R. J. Mathar, Dec 09 2016
Imaginary part of sqrt(1 + i)^3, where i is the imaginary unit such that i^2 = -1. See A154747 for real part. - Alonso del Arte, Sep 09 2019

			1.553773974030037307344158953063146948164583499410307836332671...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 7.4, p. 466.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 4.

Cf. A002965, A014176, A154747, A190283, A245592.

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

Sqrt(1+Sqrt(2)); // G. C. Greubel, Apr 14 2018

Digits:=100: evalf(sqrt(sqrt(2)+1)); # Wesley Ivan Hurt, Dec 01 2016

RealDigits[Sqrt[Sqrt[2] + 1], 10, 100][[1]] (* Wesley Ivan Hurt, Dec 01 2016 *)

sqrt(sqrt(2)+1) \\ Charles R Greathouse IV, Dec 01 2016

polrootsreal(x^4 - 2*x^2 - 1)[2] \\ Charles R Greathouse IV, Dec 01 2016

Equals 1/A154747.
Limit_{n -> oo} A002965(n)^(1/n).
From Peter Bala, Jul 01 2024: (Start)
This constant occurs in the evaluation of Integral_{x = 0..Pi/2} 1/(1 + sin^4(x)) dx = Pi/4 * sqrt(sqrt(2) + 1).
Equals 2*Sum_{n >= 0} (-1/16)^n * binomial(4*n, 2*n) (a slowly converging series). (End)
Equals 2^(3/4)*cos(Pi/8). - Vaclav Kotesovec, Jul 01 2024
Equals Product_{k>=0} coth(Pi/4 + k*Pi/2). - Antonio Graciá Llorente, Dec 19 2024
Equals sqrt(A014176) = 1/A154747 = exp(A245592). - Hugo Pfoertner, Dec 19 2024

cp's OEIS Frontend

A154747 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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A309949 Decimal expansion of the imaginary part of the square root of 1 + i.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A278928 Decimal expansion of sqrt(sqrt(2) + 1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula