

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
Named after the German astronomer and mathematician Johannes Kepler (1571  1630) and the Dutch mathematician Christoffel Jacob Bouwkamp (1915  2003).  Amiram Eldar, Aug 21 2020


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
C. J. Bouwkamp, An infinite product, Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen: Series A: Mathematical Sciences, Vol. 68 (1965), pp. 4046.
Hugo Brandt, Problem 2356, solved by Julian H. Braun, School Science and Mathematics, Vol. 53, No. 7 (1953), pp. 575576.
Marc Chamberland and Armin Straub, On gamma quotients and infinite products, Advances in Applied Mathematics, Vol. 51, No. 5 (2013), pp. 546562, preprint, arXiv:1309.3455 [math.NT], 2013. See Section 4.
Tamara Curnow, Falling down a polygonal well, Mathematical Spectrum, Vol. 26, No. 4 (1994), pp. 110118.
Tomislav 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.
Clive J. Grimstone, A product of cosines, Math. Gaz. 64 (428) (1980) 120121.
Johannes Kepler, Mysterium Cosmographicum, TÃ¼bingen, 1596. See p. 39.
M. H. Lietzke and C. W. Nestor, Jr., Problem 4793, The American Mathematical Monthly, Vol. 65, No. 6 (1958), pp. 451452, An Infinite Sequence of Inscribed Polygons, solution to Problem 4793, solved by Julian Braun and others, ibid., Vol. 66, No. 3 (1959), pp. 242243.
Kival Ngaokrajang, Illustration of polygon inscribing.
David Singmaster, Letter to the Editor: Kepler's polygonal well, Mathematical Spectrum, Vol. 27, No. 3 (1995), pp. 6364.
E. Stephens, 79.52 Slowly convergent infinite products, The Mathematical Gazette, Vol. 79, No. 486 (1995), pp. 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^(2*n)1)/n) * zeta(2*n) * (zeta(2*n)11/2^(2*n)). [Richard McIntosh]  N. J. A. Sloane, Feb 10 2008
Equals 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



