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%I #19 May 03 2018 17:37:17
%S 4,1,5,8,2,3,3,8,1,6,3,5,5,1,8,6,7,4,2,0,3,4,8,4,5,6,8,8,1,0,2,5,0,3,
%T 3,2,4,3,3,1,6,9,5,2,1,2,5,5,4,4,7,6,7,2,8,1,4,3,6,3,9,4,7,7,6,4,7,6,
%U 5,6,5,1,3,2,8,1,4,8,7,5,2,6,0,9,2,5,7,5,1,3,4,4,5,4,5,5,1,4,6,1,1,5,7,3,0
%N Decimal expansion of the edge length of a regular 15-gon with unit circumradius.
%C 15-gon is the first m-gon with odd composite m which is constructible (see A003401) in virtue of the fact that 15 is the product of two distinct Fermat primes (A019434). The next such case is 51-gon (m=3*17), followed by 85-gon (m=5*17), 771-gon (m=3*257), etc.
%C From _Wolfdieter Lang_, Apr 29 2018: (Start)
%C This constant appears in a historic problem posed by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593, solved by Viète. See the Havil reference, problem 4, pp. 69-74. See also the comments in A302711 with a link to Romanus' book, Exemplum quaesitum.
%C This problem is equivalent to R(45, 2*sin(Pi/675)) = 2*sin(Pi/15), with a special case of monic Chebyshev polynomials of the first kind, named R, given in A127672. For the constant 2*sin(Pi/675) see A302716. (End)
%D Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 69-74.
%H Stanislav Sykora, <a href="/A272534/b272534.txt">Table of n, a(n) for n = 0..2000</a>
%H Mauro Fiorentini, <a href="http://www.bitman.name/math/article/264">Construibili (numeri)</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ConstructibleNumber.html">Constructible Number</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Constructible_number">Constructible number</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Regular_polygon">Regular polygon</a>
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F Equals 2*sin(Pi/m) for m=15.
%F Also equals (sqrt(3) - sqrt(15) + sqrt(10 + 2*sqrt(5)))/4.
%F Also equals sqrt(7 - sqrt(5) - sqrt(30 - 6*sqrt(5)))/2. This is the rewritten expression of the Havil reference on top of p. 70. - _Wolfdieter Lang_, Apr 29 2018
%e 0.415823381635518674203484568810250332433169521255447672814363947...
%t RealDigits[N[2Sin[Pi/15], 100]][[1]] (* _Robert Price_, May 02 2016*)
%o (PARI) 2*sin(Pi/15)
%Y Cf. A003401, A019434, A127672, A302711, A302716.
%Y Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272535 (16), A228787 (17), A272536 (20).
%K nonn,cons,easy
%O 0,1
%A _Stanislav Sykora_, May 02 2016
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