

A121570


Decimal expansion of cosecant of 36 degrees = csc(Pi/5) = 1/sin(Pi/5).


5



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

1,2


COMMENTS

1 + csc(Pi/5) is the radius of the smallest circle into which 5 unit circles can be packed ("r=2.701+ Proved by Graham in 1968.", according to the Friedman link, which has a diagram).
csc(Pi/5) = 1/A019845 is the distance between the center of the larger circle and the center of each unit circle.
The problem of finding the diameter d of the circumscribing circle of a regular pentagon of side s = 10 (in some length units) appears as an example in Abū Kāmil's treatise on the pentagon and decagon (see the Havil reference) and Abū Kāmil links. The answer is d/s = 1/sin(Pi/5).  Wolfdieter Lang, Mar 01 2018
Longer diagonal of golden rhombus with unit edge length.  Eric W. Weisstein, Dec 11 2018


REFERENCES

Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 58.


LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000
E. Friedman, Erich's Packing Center: "Circles in Circles"
I. C. Karpinski, The Algebra of Abu Kamil, Amer. Math. Month. XXI,2 (1914), 3748.
MacTutor History of Mathematics, Abu Kamil Shuja.
Eric Weisstein's World of Mathematics, Golden Rhombus
Wikipedia, Abu Kamil.


FORMULA

1/sin(Pi/5) = 2*(2*phi  1)*sqrt(2 + phi)/5, with the golden ratio phi = A001622.  Wolfdieter Lang, Mar 01 2018


EXAMPLE

1.701301616704079864363080994126...


MAPLE

evalf(1/sin(Pi/5), 130); # Muniru A Asiru, Nov 02 2018


MATHEMATICA

RealDigits[Csc[Pi/5], 10, 100][[1]] (* G. C. Greubel, Nov 02 2018 *)


PROG

(PARI) 1/sin(Pi/5)
(MAGMA) SetDefaultRealField(RealField(100)); R:= RealField(); 1/Sin(Pi(R)/5); // G. C. Greubel, Nov 02 2018
(Sage) numerical_approx(1/sin(pi/5), digits=100) # G. C. Greubel, Dec 12 2018


CROSSREFS

Cf. A001622, A019845 (1/A121570), A182007 (2/A121570).
Cf. A179290 (shorter golden rhombus diagonal).
Sequence in context: A061846 A293530 A199603 * A169681 A202354 A324498
Adjacent sequences: A121567 A121568 A121569 * A121571 A121572 A121573


KEYWORD

cons,nonn


AUTHOR

Rick L. Shepherd, Aug 08 2006


STATUS

approved



