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A085365 Decimal expansion of the Kepler-Bouwkamp or polygon-inscribing constant. 4
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, 6, 5, 6, 8, 1, 2, 6, 8, 4, 7, 5, 3, 6, 0, 0, 4, 3, 1, 4, 8, 4, 7, 3, 4, 2, 5, 9, 7, 6, 4, 2, 8, 2, 3, 8, 1, 6, 5, 8, 9, 2, 4, 4, 1, 2, 7, 0, 8, 8, 1, 7, 3, 0, 8, 7, 3, 3, 8, 9, 7, 7, 7, 2, 4, 5, 6, 2, 7, 6, 5, 0, 5, 5, 1, 0, 0, 3, 7 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

"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

REFERENCES

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

S. R. Finch, Mathematical Constants. Cambridge University Press (2003). MR 2003519.

LINKS

Table of n, a(n) for n=0..104.

M. Chamberland, A. Straub, On Gamma quotients and infinite products, arXiv:1309.3455, Section 4.

Clive J. Grimstone, A product of cosines, Math. Gaz. 64 (428) (1980) 120-121.

Kival Ngaokrajang, Illustration of polygon inscribing

R. Stephens, Slowly converging infinite products, Math. Gaz. 79 (486) (1995) 561-565.

Eric Weisstein's World of Mathematics, Polygon Inscribing

Wikipedia, Kepler-Bouwkamp constant

FORMULA

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

A085365 = 1/A051762. - M. F. Hasler, May 18 2014

EXAMPLE

0.1149420448532...

MATHEMATICA

RealDigits[ N[ Product[ Cos[ Pi/n], {n, 3, Infinity}], 111]][[1]] (* Robert G. Wilson v, May 22 2014 *)

PROG

(PARI) exp(sumpos(n=3, log(cos(Pi/n)))) \\ M. F. Hasler, May 18 2014

CROSSREFS

Equals 1/A051762.

Cf. A131671.

Sequence in context: A143298 A177839 A013669 * A019767 A244994 A021091

Adjacent sequences:  A085362 A085363 A085364 * A085366 A085367 A085368

KEYWORD

nonn,cons

AUTHOR

Eric W. Weisstein, Jun 25 2003

EXTENSIONS

Edited by M. F. Hasler, May 18 2014

Corrected and extended by Robert G. Wilson v, May 22 2014

STATUS

approved

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Last modified July 29 14:58 EDT 2014. Contains 245039 sequences.