A085365 - OEIS

%I #76 Feb 16 2025 08:32:50

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

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

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

%N Decimal expansion of the Kepler-Bouwkamp or polygon-inscribing constant.

%C Inscribe an equilateral triangle in a circle of unit radius. Inscribe a circle in the triangle. Inscribe a square in the second circle and inscribe a circle in the square. Inscribe a regular pentagon in the third circle and so on. The radii of the circles converge to Product_{ k = 3..infinity } cos(Pi/k), which is this number. - _N. J. A. Sloane_, Feb 10 2008

%C "It is stated in Kasner and Newman's 'Mathematics and the Imagination' (pp. 269-270 in the Pelican edition) that P=Product{k=3..infinity} cos(Pi/k) is approximately equal to 1/12. Not so! ..., so that a very good approximation to P is 10/87 ...", by Grimstone. - _Robert G. Wilson v_, Dec 22 2013

%C Named after the German astronomer and mathematician Johannes Kepler (1571 - 1630) and the Dutch mathematician Christoffel Jacob Bouwkamp (1915 - 2003). - _Amiram Eldar_, Aug 21 2020

%D Dick Katz, Problem 91:24, in R. K. Guy, ed., Western Number Theory Problems, 1992-12-19 & 22.

%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, p. 382.

%H Vaclav Kotesovec, <a href="/A085365/b085365.txt">Table of n, a(n) for n = 0..1000</a>

%H C. J. Bouwkamp, <a href="https://core.ac.uk/download/pdf/82376060.pdf">An infinite product</a>, Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen: Series A: Mathematical Sciences, Vol. 68 (1965), pp. 40-46.

%H Hugo Brandt, <a href="https://doi.org/10.1111/j.1949-8594.1953.tb06870.x">Problem 2356</a>, solved by Julian H. Braun, School Science and Mathematics, Vol. 53, No. 7 (1953), pp. 575-576.

%H Marc Chamberland and Armin Straub, <a href="https://doi.org/10.1016/j.aam.2013.07.003">On gamma quotients and infinite products</a>, Advances in Applied Mathematics, Vol. 51, No. 5 (2013), pp. 546-562, <a href="http://arxiv.org/abs/1309.3455">preprint</a>, arXiv:1309.3455 [math.NT], 2013. See Section 4.

%H Tamara Curnow, <a href="https://drive.google.com/file/d/12Vf9pCD_AQuOTCswx7af-kEdokLHNYoW/view">Falling down a polygonal well</a>, Mathematical Spectrum, Vol. 26, No. 4 (1994), pp. 110-118.

%H Tomislav Doslic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Doslic/doslic3.html">Kepler-Bouwkamp Radius of Combinatorial Sequences</a>, J. Int. Seq. 17 (2014) # 14.11.3.

%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, p. 58.

%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 Johannes Kepler, <a href="https://archive.org/details/1596-kepler-prodromus-dissertationum-cosmographicarum-continens-mysterium-cosmographicum/page/38/mode/2up">Mysterium Cosmographicum</a>, Tübingen, 1596. See p. 39.

%H M. H. Lietzke and C. W. Nestor, Jr., <a href="https://www.jstor.org/stable/2310734">Problem 4793</a>, The American Mathematical Monthly, Vol. 65, No. 6 (1958), pp. 451-452, <a href="https://www.jstor.org/stable/2309535">An Infinite Sequence of Inscribed Polygons, solution to Problem 4793</a>, solved by Julian Braun and others, ibid., Vol. 66, No. 3 (1959), pp. 242-243.

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

%H David Singmaster, <a href="http://www.appliedprobability.org/content.aspx?Group=ms&amp;Page=MS273">Letter to the Editor: Kepler's polygonal well</a>, Mathematical Spectrum, Vol. 27, No. 3 (1995), pp. 63-64.

%H E. Stephens, <a href="https://www.jstor.org/stable/3618092">79.52 Slowly convergent infinite products</a>, The Mathematical Gazette, Vol. 79, No. 486 (1995), pp. 561-565.

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

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Kepler-Bouwkamp_constant">Kepler-Bouwkamp constant</a>.

%F Equals Product_{n>=3} cos(Pi/n).

%F The log of this constant is equal to Sum_{n=1..infinity} -((2^(2*n)-1)/n) * zeta(2*n) * (zeta(2*n)-1-1/2^(2*n)). [Richard McIntosh] - _N. J. A. Sloane_, Feb 10 2008

%F Equals 1/A051762. - _M. F. Hasler_, May 18 2014

%F Equals product A365255 * A365256. - _R. J. Mathar_, Aug 30 2023

%e 0.1149420448532...

%p evalf(1/(product(sec(Pi/k), k=3..infinity)), 104) # _Vaclav Kotesovec_, Sep 20 2014

%t (* The naive approach, N[ Product[ Cos[ Pi/n], {n, 3, Infinity}], 111], yields only 27 correct decimals. - _Vaclav Kotesovec_, Sep 20 2014 *)

%t Block[{$MaxExtraPrecision = 1000}, Do[Print[N[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)))) \\ _M. F. Hasler_, May 18 2014

%Y Cf. A051762, A131671, A365255, A365256.

%K nonn,cons

%O 0,3

%A _Eric W. Weisstein_, Jun 25 2003

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

%E First formula corrected (missing sign) by _Vaclav Kotesovec_, Sep 20 2014

%E Terms since 27 corrected by _Vaclav Kotesovec_, Sep 20 2014 (recomputed with higher precision)