OFFSET
0,1
COMMENTS
Constant related to the constructibility of the regular heptadecagon. The "Disquisitiones Arithmeticae" of Gauss contains the following equivalent expression:
-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)))).
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.
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,...
REFERENCES
C. F. Gauss, Disquisitiones Arithmeticae, 1801 (Lipsia), p. 662 (par. 365).
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.
Ian Stewart, Why Beauty Is Truth, A History of Symmetry, BASIC Books, a member of the Perseus Books Group, NY 2007, pp. 136.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Heptadecagon.
Wikipedia, Heptadecagon.
FORMULA
Equals (i^(4/17) - i^(30/17))/2. - Peter Luschny, Apr 04 2020
EXAMPLE
cos(2*Pi/17) = 0.9324722294043558045731158918215633862625877779451169...
MATHEMATICA
RealDigits[Cos[2Pi/17], 10, 105][[1]]
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 *)
PROG
(PARI) cos(2*Pi/17)
(Maxima) fpprec:90; ev(bfloat(cos(2*%pi/17)));
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Bruno Berselli, Mar 26 2012
STATUS
approved