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%I #41 Mar 13 2024 05:11:04
%S 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,
%T 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,
%U 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
%N Decimal expansion of cosecant of 22.5 degrees = csc(Pi/8).
%C 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).
%C 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.
%C 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
%C 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
%D D. Mumford et al., Indra's Pearls, Cambridge 2002; see p. 362. - _N. J. A. Sloane_, Nov 22 2009
%H G. C. Greubel, <a href="/A121601/b121601.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 2*sqrt(2)*cos(Pi/8).
%F Equals Product_{k >= 0} (8*k + 4)^2/((8*k + 1)*(8*k + 7)). - _Antonio Graciá Llorente_, Mar 11 2024
%e 2.6131259297527530557132863468543743071675223766985390550977...
%p evalf(1/sin(Pi/8),120); # _Muniru A Asiru_, Nov 02 2018
%t RealDigits[Csc[Pi/8],10,130][[1]] (* corrected by _Harvey P. Dale_, Jul 28 2012 *)
%o (PARI) 1/sin(Pi/8)
%o (Magma) SetDefaultRealField(RealField(100)); R:=RealField(); 1/Sin(Pi(R)/8); // _G. C. Greubel_, Nov 02 2018
%Y Cf. A121598, A179260.
%K cons,nonn
%O 1,1
%A _Rick L. Shepherd_, Aug 09 2006
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