

A051762


Polygon circumscribing constant: decimal expansion of Product_{n=3..infinity} 1/cos(Pi/n).


8



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.


LINKS

Table of n, a(n) for n=1..105.
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) 120121.
A. R. Kitson, The prime analog of the KeplerBouwkamp constant, arXiv:math/0608186.
R. J. Mathar, Tightly circumscribed regular polygons, arXiv:1301.6293 [math.MG]
Kival Ngaokrajang, Illustration of polygon inscribing
Eric Weisstein's World of Mathematics, Polygon Circumscribing
Wikipedia, Polygon circumscribing constant


FORMULA

A051762 = 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

Cf. A085365, A118253, A131671, A211174.
Sequence in context: A112145 A248291 A038284 * A247017 A198112 A213007
Adjacent sequences: A051759 A051760 A051761 * A051763 A051764 A051765


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



