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A121570
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Decimal expansion of cosecant of 36 degrees = csc(Pi/5) = 1/sin(Pi/5).
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5
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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
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1,2
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COMMENTS
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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
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REFERENCES
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Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 58.
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LINKS
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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), 37-48.
MacTutor History of Mathematics, Abu Kamil Shuja.
Eric Weisstein's World of Mathematics, Golden Rhombus
Wikipedia, Abu Kamil.
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FORMULA
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1/sin(Pi/5) = 2*(2*phi - 1)*sqrt(2 + phi)/5, with the golden ratio phi = A001622. - Wolfdieter Lang, Mar 01 2018
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EXAMPLE
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1.701301616704079864363080994126...
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MAPLE
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evalf(1/sin(Pi/5), 130); # Muniru A Asiru, Nov 02 2018
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MATHEMATICA
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RealDigits[Csc[Pi/5], 10, 100][[1]] (* G. C. Greubel, Nov 02 2018 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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cons,nonn
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AUTHOR
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Rick L. Shepherd, Aug 08 2006
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STATUS
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approved
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