A190283 -id:A190283 - cp's OEIS frontend

A190284 Continued fraction of 1+sqrt(1+sqrt(2)).

2, 1, 1, 4, 6, 1, 2, 2, 2, 1, 1, 6, 1, 179, 46, 1, 1, 3, 2, 1, 1, 3, 6, 3, 1, 1, 1, 1, 2, 1, 1, 56, 1, 1, 1, 1, 66, 1, 1, 2, 17, 8, 2, 7, 12, 1, 1, 8, 1, 2, 2, 1, 1, 2, 1, 12, 1, 2, 2, 2, 2, 1, 1, 1, 8, 1, 1, 1, 1, 2, 1, 2, 5, 1, 6, 8, 1, 1, 1, 2, 7, 1, 9, 1, 2, 5, 7, 1, 6, 1, 10, 1, 2, 1, 3, 47, 1, 1, 998, 1

Offset: 1

Clark Kimberling, May 07 2011

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A188635, A190283, A190282, A154748.

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

FromContinuedFraction[{2, Sqrt[2], {2, r}}]
FullSimplify[%]
ContinuedFraction[%, 100]  (* A190284 *)
RealDigits[N[%%, 120]]     (* A190283 *)

contfrac(1+sqrt(1+sqrt(2))) \\ G. C. Greubel, Apr 14 2018

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

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A190284 Continued fraction of 1+sqrt(1+sqrt(2)).

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

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