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

1,1

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

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.

Formula

Equals 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,changed

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