A285871 Decimal expansion of 1/sqrt(2 - sqrt(2)) (reciprocal of A101464).

1, 3, 0, 6, 5, 6, 2, 9, 6, 4, 8, 7, 6, 3, 7, 6, 5, 2, 7, 8, 5, 6, 6, 4, 3, 1, 7, 3, 4, 2, 7, 1, 8, 7, 1, 5, 3, 5, 8, 3, 7, 6, 1, 1, 8, 8, 3, 4, 9, 2, 6, 9, 5, 2, 7, 5, 4, 8, 8, 9, 8, 3, 6, 6, 9, 0, 8, 0, 8, 1, 0, 4, 1, 4, 6, 1, 1, 9, 2, 0, 5, 0, 9, 5, 1, 8, 5, 3, 7, 2, 0, 1, 9, 2, 6, 2, 8, 1, 4

Offset: 1

Author

Wolfdieter Lang, May 11 2017

Keywords

• nonn
• cons
• easy

Comments

This number is the length ratio of the radius of a circle and the side of the inscribed octagon.

In the Corbalán reference, pp. 61-62, this number is called Cordoba number or Cordoba proportion, attributed to the architect Rafael de la Hoz (1924-2000), who used the rectangle with this proportion to explain the structure of the Mihrab of Cordoba.

Examples

			1.30656296487637652785664317342718715358376118834926952754889836690808104146...
		

References

• Fernando Corbalán, Der goldene Schnitt, Librero, 2017. Original: La proportión áurea, RBA Contenidos Editoriales y Audiovisuales S. A. U., 2010. English: The golden Ratio, 2012, RBA Coleccionables.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10001 [offset adapted to 1 by _Georg Fischer_, Sep 03 2020]
• Eric Weisstein's World of Mathematics, Octagon.
• Wikipedia, Octagon.
• Index entries for algebraic numbers, degree 4.

Crossrefs

Cf. A101464, A179260.

Programs

Magma

SetDefaultRealField(RealField(100)); 1/Sqrt(2 - Sqrt(2)); // G. C. Greubel, Oct 10 2018

Maple

evalf((sqrt(2-sqrt(2)))^(-1),100); # Muniru A Asiru, Oct 11 2018

Mathematica

RealDigits[1/Sqrt[2 - Sqrt[2]], 10, 100][[1]] (* Indranil Ghosh, May 11 2017 *)

PARI

default(realprecision, 100); 1/sqrt(2 - sqrt(2)) \\ G. C. Greubel, Oct 10 2018

Python

from sympy import N, sqrt
print(N(1/sqrt(2 - sqrt(2)), 100)) # Indranil Ghosh, May 11 2017

Formula

Equals 1/(2*sin(Pi/8)) = 1/A101464.

Equals Product_{k>=0} (1 + (-1)^k/(4*k+2)). - Amiram Eldar, Aug 07 2020

The minimal polynomial is 2*x^4 - 4*x^2 + 1. - Joerg Arndt, May 10 2021

Equals Sum_{n>=0} binomial(2*n - 1/2, -1/2)/2^n. - Antonio Graciá Llorente, Nov 13 2024

Extensions

Offset and example corrected by Amiram Eldar, Aug 07 2020