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 A051762 Polygon circumscribing constant: decimal expansion of Product_{n=3..infinity} 1/cos(Pi/n). 11
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. 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 REFERENCES 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. 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. LINKS M. Chamberland, A. Straub, On Gamma quotients and infinite products, arXiv:1309.3455 [math.NT], 2013, Section 4. Clive J. Grimstone, A product of cosines, Math. Gaz. 64 (428) (1980) 120-121. A. R. Kitson, The prime analog of the Kepler-Bouwkamp constant, arXiv:math/0608186 [math.HO], 2006. R. J. Mathar, Tightly circumscribed regular polygons, arXiv:1301.6293 [math.MG], 2013. Kival Ngaokrajang, Illustration of polygon inscribing Eric Weisstein's World of Mathematics, Polygon Circumscribing Wikipedia, Polygon circumscribing constant FORMULA A051762 = 1/A085365. EXAMPLE 8.700036625208194503222409859113004971193297949742892092159667278683429964114... MAPLE evalf(product(sec(Pi/k), k=3..infinity), 103) # Vaclav Kotesovec, Sep 20 2014 MATHEMATICA (* 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] 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 *) PROG (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 CROSSREFS Cf. A085365, A118253, A131671, A211174. Sequence in context: A038284 A264587 A265024 * A247017 A330142 A198112 Adjacent sequences:  A051759 A051760 A051761 * A051763 A051764 A051765 KEYWORD nonn,cons AUTHOR Robert G. Wilson v, Aug 23 2000 EXTENSIONS More terms from Eric W. Weisstein, Jun 25 2003 Edited by M. F. Hasler, May 18 2014 Example corrected by Vaclav Kotesovec, Sep 20 2014 STATUS approved

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Last modified June 29 18:25 EDT 2022. Contains 354913 sequences. (Running on oeis4.)