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

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

1,2

Comments

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

References

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Index entries for algebraic numbers, degree 4.

Formula

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

Example

1.553773974030037307344158953063146948164583499410307836332671...

Maple

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

Mathematica

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

Prog

(PARI) sqrt(sqrt(2)+1) \\ Charles R Greathouse IV, Dec 01 2016
(PARI) polrootsreal(x^4 - 2*x^2 - 1)[2] \\ Charles R Greathouse IV, Dec 01 2016
(Magma) Sqrt(1+Sqrt(2)); // G. C. Greubel, Apr 14 2018

Crossrefs

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

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

Sequence in context: A232813 A267033 A306982 * A273826 A213054 A232609

Adjacent sequences: A278925 A278926 A278927 * A278929 A278930 A278931

Keyword

nonn,cons,changed

Author

Bobby Jacobs, Dec 01 2016

Status

approved