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

Author

Bobby Jacobs, Dec 01 2016

Keywords

• nonn
• cons

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

Examples

			1.553773974030037307344158953063146948164583499410307836332671...
		

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.

Crossrefs

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

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

Programs

Magma

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

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 *)

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

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