

A085365


Decimal expansion of the KeplerBouwkamp or polygoninscribing constant.


9



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, 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, 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
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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. 269270 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, 19921219 & 22.
S. R. Finch, Mathematical Constants. Cambridge University Press (2003). MR 2003519.
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.


LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
M. Chamberland, A. Straub, On Gamma quotients and infinite products, arXiv:1309.3455, Section 4.
T. Doslic, KeplerBouwkamp Radius of Combinatorial Sequences, J. Int. Seq. 17 (2014) # 14.11.3
Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 58.
Steven R. Finch, Errata and Addenda to Mathematical Constants, January 22, 2016. [Cached copy, with permission of the author]
Clive J. Grimstone, A product of cosines, Math. Gaz. 64 (428) (1980) 120121.
Kival Ngaokrajang, Illustration of polygon inscribing
R. Stephens, Slowly converging infinite products, Math. Gaz. 79 (486) (1995) 561565.
Eric Weisstein's World of Mathematics, Polygon Inscribing
Wikipedia, KeplerBouwkamp constant


FORMULA

The log of this constant is equal to Sum_{n=1..infinity} ((2^(2n)1)/n) * zeta(2n) * (zeta(2n)11/2^(2n)). [Richard McIntosh]  N. J. A. Sloane, Feb 10 2008
A085365 = 1/A051762.  M. F. Hasler, May 18 2014


EXAMPLE

0.1149420448532...


MAPLE

evalf(1/(product(sec(Pi/k), k=3..infinity)), 104) # Vaclav Kotesovec, Sep 20 2014


MATHEMATICA

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


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
First formula corrected (missing sign) by Vaclav Kotesovec, Sep 20 2014
Terms since 27 corrected by Vaclav Kotesovec, Sep 20 2014 (recomputed with higher precision)


STATUS

approved



