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A085365
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Decimal expansion of the Kepler-Bouwkamp or polygon-inscribing constant.
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11
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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
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0,3
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COMMENTS
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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
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
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REFERENCES
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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.
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.
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LINKS
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C. J. Bouwkamp, An infinite product, Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen: Series A: Mathematical Sciences, Vol. 68 (1965), pp. 40-46.
Hugo Brandt, Problem 2356, solved by Julian H. Braun, School Science and Mathematics, Vol. 53, No. 7 (1953), pp. 575-576.
M. H. Lietzke and C. W. Nestor, Jr., Problem 4793, The American Mathematical Monthly, Vol. 65, No. 6 (1958), pp. 451-452, An Infinite Sequence of Inscribed Polygons, solution to Problem 4793, solved by Julian Braun and others, ibid., Vol. 66, No. 3 (1959), pp. 242-243.
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FORMULA
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Equals Product_{n>=3} cos(Pi/n).
The log of this constant is equal to Sum_{n=1..infinity} -((2^(2*n)-1)/n) * zeta(2*n) * (zeta(2*n)-1-1/2^(2*n)). [Richard McIntosh] - N. J. A. Sloane, Feb 10 2008
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EXAMPLE
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0.1149420448532...
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MAPLE
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evalf(1/(product(sec(Pi/k), k=3..infinity)), 104) # Vaclav Kotesovec, Sep 20 2014
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MATHEMATICA
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(* 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 *)
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PROG
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(PARI) exp(sumpos(n=3, log(cos(Pi/n)))) \\ M. F. Hasler, May 18 2014
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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Terms since 27 corrected by Vaclav Kotesovec, Sep 20 2014 (recomputed with higher precision)
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STATUS
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approved
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