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A278928
Decimal expansion of sqrt(sqrt(2) + 1).
2
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
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
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. A309948 and A309949 for real and imaginary parts of sqrt(1 + i).
Sequence in context: A232813 A267033 A306982 * A273826 A213054 A232609
KEYWORD
nonn,cons
AUTHOR
Bobby Jacobs, Dec 01 2016
STATUS
approved