A210644 Decimal expansion of cos(2*Pi/17).

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

Offset: 0

Author

Bruno Berselli, Mar 26 2012

Keywords

• nonn
• cons

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,...

Examples

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

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
• Brady Haran and David Eisenbud, Heptadecagon and Fermat Primes (the math bit), Numberphile YouTube video, 2015.
• Eric Weisstein's World of Mathematics, Heptadecagon.
• Wikipedia, Heptadecagon.
• Index entries for algebraic numbers, degree 8.

Crossrefs

Cf. A019700, A210649.

Programs

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 *)

Maxima

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

PARI

cos(2*Pi/17)

Formula

Equals (i^(4/17) - i^(30/17))/2. - Peter Luschny, Apr 04 2020