A121601 Decimal expansion of cosecant of 22.5 degrees = csc(Pi/8).

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

Offset: 1

Author

Rick L. Shepherd, Aug 09 2006

Keywords

• cons
• nonn

Comments

1 + csc(Pi/8) is the radius of the smallest circle into which 9 unit circles can be packed ("r=3.613+ Proved by Pirl in 1969", according to the Friedman link, which has a diagram).

csc(Pi/8) is the distance between the center of the larger circle and the center of each unit circle that touches the larger circle.

A rectangle of length L and width W is a called a silver rectangle if L=rW, where r is the silver ratio; i.e., r = 1+sqrt(2). The diagonal has length D = sqrt(W^2+L^2), so that D/W = sqrt(4+2*sqrt(2)) = csc(Pi/8). - Clark Kimberling, Apr 04 2011

This algebraic integer of degree 4 also gives the length ratio diagonal/side of the longest diagonal in the regular octagon. The minimal polynomial is x^4 - 8*x + 8. In the power basis of Gal(Q(rho(8))/Q), with rho(8) = sqrt(2 + sqrt(2)) = A179260 it is -2*rho(8) + 1*rho(8)^3 which equals sqrt(2)*rho(8). - Wolfdieter Lang, Oct 28 2020

Examples

			2.6131259297527530557132863468543743071675223766985390550977...
		

References

• D. Mumford et al., Indra's Pearls, Cambridge 2002; see p. 362. - N. J. A. Sloane, Nov 22 2009

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Index entries for algebraic numbers, degree 4

Crossrefs

Cf. A121598, A179260.

Programs

Magma

SetDefaultRealField(RealField(100)); R:=RealField(); 1/Sin(Pi(R)/8); // G. C. Greubel, Nov 02 2018

Maple

evalf(1/sin(Pi/8),120); # Muniru A Asiru, Nov 02 2018

Mathematica

RealDigits[Csc[Pi/8],10,130][[1]] (* corrected by Harvey P. Dale, Jul 28 2012 *)

PARI

1/sin(Pi/8)

Formula

Equals 2*sqrt(2)*cos(Pi/8).

Equals Product_{k >= 0} (8*k + 4)^2/((8*k + 1)*(8*k + 7)). - Antonio Graciá Llorente, Mar 11 2024