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A051762 Polygon circumscribing constant: decimal expansion of Product_{n>=3} 1/cos(Pi/n). 11
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 (list; constant; graph; refs; listen; history; text; internal format)
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
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
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.
Eric Weisstein's World of Mathematics, Polygon Circumscribing
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
Sequence in context: A038284 A264587 A265024 * A247017 A330142 A198112
KEYWORD
nonn,cons
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

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)