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%I #54 Jul 01 2024 13:16:13
%S 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,
%T 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,
%U 7,5,2,5,6,7,8,8,7,3,3,1,0,2,7,2,7,9
%N Decimal expansion of sqrt(sqrt(2) + 1).
%C A quartic integer with minimal polynomial x^4 - 2*x^2 - 1. - _Charles R Greathouse IV_, Dec 01 2016
%C 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.
%C Apart from the first digit, the same as A190283. - _R. J. Mathar_, Dec 09 2016
%C 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
%H G. C. Greubel, <a href="/A278928/b278928.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>
%F Equals 1/A154747.
%F Limit_{n -> oo} A002965(n)^(1/n).
%F From _Peter Bala_, Jul 01 2024: (Start)
%F This constant occurs in the evaluation of Integral_{x = 0..Pi/2} 1/(1 + sin^4(x)) dx = Pi/4 * sqrt(sqrt(2) + 1).
%F Equals 2*Sum_{n >= 0} (-1/16)^n * binomial(4*n, 2*n) (a slowly converging series). (End)
%F Equals 2^(3/4)*cos(Pi/8). - _Vaclav Kotesovec_, Jul 01 2024
%e 1.553773974030037307344158953063146948164583499410307836332671...
%p Digits:=100: evalf(sqrt(sqrt(2)+1)); # _Wesley Ivan Hurt_, Dec 01 2016
%t RealDigits[Sqrt[Sqrt[2] + 1], 10, 100][[1]] (* _Wesley Ivan Hurt_, Dec 01 2016 *)
%o (PARI) sqrt(sqrt(2)+1) \\ _Charles R Greathouse IV_, Dec 01 2016
%o (PARI) polrootsreal(x^4 - 2*x^2 - 1)[2] \\ _Charles R Greathouse IV_, Dec 01 2016
%o (Magma) Sqrt(1+Sqrt(2)); // _G. C. Greubel_, Apr 14 2018
%Y Cf. A002965, A154747, A190283.
%Y Cf. A309948 and A309949 for real and imaginary parts of sqrt(1 + i).
%K nonn,cons
%O 1,2
%A _Bobby Jacobs_, Dec 01 2016
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