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A121570
Decimal expansion of cosecant of 36 degrees = csc(Pi/5) = 1/sin(Pi/5).
9
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
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
The length of the longer side of a golden rectangle inscribed in a unit circle. - Michal Paulovic, Sep 01 2022
The radius of a common circle surrounded by 5 tangent unit circles is A121570 - 1. - Thomas Otten, Dec 27 2023
REFERENCES
Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 58.
LINKS
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.
FORMULA
Equals 1/A019845.
Equals 2*(2*phi - 1)*sqrt(2 + phi)/5, with the golden ratio phi = A001622. - Wolfdieter Lang, Mar 01 2018
Equals sqrt(2 + 2 / sqrt(5)). - Michal Paulovic, Sep 01 2022
The minimal polynomial is 5*x^4 - 20*x^2 + 16. - Joerg Arndt, Sep 09 2022
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 (inverse), A182007 (2/A121570).
Cf. A179290 (shorter golden rhombus diagonal).
Sequence in context: A335947 A293530 A199603 * A169681 A202354 A324498
KEYWORD
cons,nonn
AUTHOR
Rick L. Shepherd, Aug 08 2006
STATUS
approved