A051762 Polygon circumscribing constant: decimal expansion of Product_{n>=3} 1/cos(Pi/n).

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

Offset: 1

Author

Robert G. Wilson v, Aug 23 2000

Keywords

• nonn
• cons

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

Examples

			8.700036625208194503222409859113004971193297949742892092159667278683429964114...
		

References

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

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000
• M. Chamberland and 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.

Crossrefs

Cf. A085365, A118253, A131671, A211174.

Programs

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 *)

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

Formula

Equals 1/A085365.

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