A210644 - OEIS

%I #46 Jan 18 2025 15:43:10

%S 9,3,2,4,7,2,2,2,9,4,0,4,3,5,5,8,0,4,5,7,3,1,1,5,8,9,1,8,2,1,5,6,3,3,

%T 8,6,2,6,2,5,8,7,7,7,7,9,4,5,1,1,6,9,2,8,2,4,8,3,5,0,0,1,1,8,6,0,5,3,

%U 6,0,4,6,5,6,9,6,4,4,4,9,8,1,2,8,0,7,4

%N Decimal expansion of cos(2*Pi/17).

%C Constant related to the constructibility of the regular heptadecagon. The "Disquisitiones Arithmeticae" of Gauss contains the following equivalent expression:

%C -1/16+(1/16)*sqrt(17)+(1/16)*sqrt(34-2*sqrt(17))+(1/8)*sqrt(17+3*sqrt(17)-sqrt(34-2*sqrt(17))-2*sqrt((34+2*sqrt(17)))).

%C This value is a root of the polynomial 256*x^8+128*x^7-448*x^6-192*x^5+240*x^4+80*x^3-40*x^2-8*x+1.

%C The continued fraction expansion of cos(2*Pi/17) is 0, 1, 13, 1, 4, 4, 2, 1, 1, 2, 4, 425, 1, 2, 5, 3, 1, 1, 1, 1, 1, 4, 4, 10, 3, 2, 1,...

%D C. F. Gauss, Disquisitiones Arithmeticae, 1801 (Lipsia), p. 662 (par. 365).

%D Ian Stewart, Professor Stewart's Cabinet of Mathematical Curiosities, BASIC Books, a member of the Perseus Books Group, NY, 2009, "Why Gauss Became a Mathematician", pp. 146 - 149.

%D Ian Stewart, Why Beauty Is Truth, A History of Symmetry, BASIC Books, a member of the Perseus Books Group, NY 2007, pp. 136.

%H Vincenzo Librandi, <a href="/A210644/b210644.txt">Table of n, a(n) for n = 0..1000</a>

%H Brady Haran and David Eisenbud, <a href="https://www.youtube.com/watch?v=oYlB5lUGlbw">Heptadecagon and Fermat Primes (the math bit)</a>, Numberphile YouTube video, 2015.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Heptadecagon.html">Heptadecagon</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Heptadecagon">Heptadecagon</a>.

%F Equals (i^(4/17) - i^(30/17))/2. - _Peter Luschny_, Apr 04 2020

%e cos(2*Pi/17) = 0.9324722294043558045731158918215633862625877779451169...

%t RealDigits[Cos[2Pi/17], 10, 105][[1]]

%t RealDigits[(-1 + Sqrt[17] + Sqrt[34 - 2 Sqrt[17]] + Sqrt[68 + 12 Sqrt[17] - 4 Sqrt[170 + 38 Sqrt[17]]])/16, 10, 111][[1]] (* _Robert G. Wilson v_, Aug 09 2012 *)

%o (PARI) cos(2*Pi/17)

%o (Maxima) fpprec:90; ev(bfloat(cos(2*%pi/17)));

%Y Cf. A019700, A210649.

%K nonn,cons,changed

%O 0,1

%A _Bruno Berselli_, Mar 26 2012