A051762 - OEIS

%I #78 Feb 16 2025 08:32:41

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

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

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

%N Polygon circumscribing constant: decimal expansion of Product_{n>=3} 1/cos(Pi/n).

%C The geometric interpretation is as follows. Begin with a unit circle. Circumscribe an equilateral triangle and then circumscribe a circle. Circumscribe a square and then circumscribe a circle. Circumscribe a regular pentagon and then circumscribe a circle, etc. The circles have radius which converges to this value.

%C Grimstone corrects an error in other references and gives an approximation for 1/A085365, see there for further information. - _M. F. Hasler_, May 18 2014

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.3, p. 428.

%D Clifford A. Pickover, The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publ., NY, 2009, page 382.

%D A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 1 (Overseas Publishers Association, Amsterdam, 1986), p. 757, section 6.2.4, formula 1.

%H Robert G. Wilson v, <a href="/A051762/b051762.txt">Table of n, a(n) for n = 1..10000</a>

%H M. Chamberland and A. Straub, <a href="https://arxiv.org/abs/1309.3455">On Gamma quotients and infinite products</a>, arXiv:1309.3455 [math.NT], 2013, Section 4.

%H Clive J. Grimstone, <a href="http://www.jstor.org/stable/3615085">A product of cosines</a>, Math. Gaz. 64 (428) (1980) 120-121.

%H A. R. Kitson, <a href="https://arxiv.org/abs/math/0608186">The prime analog of the Kepler-Bouwkamp constant</a>, arXiv:math/0608186 [math.HO], 2006.

%H R. J. Mathar, <a href="https://arxiv.org/abs/1301.6293">Tightly circumscribed regular polygons</a>, arXiv:1301.6293 [math.MG], 2013.

%H Kival Ngaokrajang, <a href="/A085365/a085365.jpg">Illustration of polygon inscribing</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PolygonCircumscribing.html">Polygon Circumscribing</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Polygon_circumscribing_constant">Polygon circumscribing constant</a>.

%F Equals 1/A085365.

%e 8.700036625208194503222409859113004971193297949742892092159667278683429964114...

%p evalf(product(sec(Pi/k), k=3..infinity), 103) # _Vaclav Kotesovec_, Sep 20 2014

%t (* A check of the calculation can be made by dividing the product into two halves, a = N[Product[1/Cos[Pi/(2 n + 1)], {n, 1, Infinity}],111], b = N[Product[1/Cos[Pi/(2 n)], {n, 2, Infinity}],111] and a*b = A051762. - Robert G. Wilson v, Dec 22 2013 *) [This approach turns out to give incorrect numerical results. - _M. F. Hasler_, Sep 20 2014]

%t Block[{$MaxExtraPrecision = 1000}, Do[Print[N[1/Exp[Sum[-(2^(2*n)-1)/n * Zeta[2*n]*(Zeta[2*n] - 1 - 1/2^(2*n)), {n, 1, m}]], 110]], {m, 250, 300}]] (* over 100 decimal places are correct, _Vaclav Kotesovec_, Sep 20 2014 *)

%o (PARI) exp(-sumpos(n=3,log(cos(Pi/n)))) \\ Converges very quickly, which is not the case for suminf(...) or prodinf(cos(Pi/n)). \\ _M. F. Hasler_, May 18 2014

%Y Cf. A085365, A118253, A131671, A211174.

%K nonn,cons

%O 1,1

%A _Robert G. Wilson v_, Aug 23 2000

%E More terms from _Eric W. Weisstein_, Jun 25 2003

%E Edited by _M. F. Hasler_, May 18 2014

%E Example corrected by _Vaclav Kotesovec_, Sep 20 2014