

A090986


Decimal expansion of Pi/sinh(Pi).


16



2, 7, 2, 0, 2, 9, 0, 5, 4, 9, 8, 2, 1, 3, 3, 1, 6, 2, 9, 5, 0, 2, 3, 6, 5, 8, 3, 6, 7, 2, 0, 3, 7, 5, 5, 5, 8, 4, 0, 7, 1, 8, 3, 6, 3, 4, 6, 0, 3, 1, 5, 9, 4, 9, 5, 0, 6, 8, 9, 6, 7, 8, 3, 8, 5, 6, 2, 4, 6, 1, 9, 1, 3, 6, 9, 4, 8, 7, 8, 8, 8, 1, 9, 1, 1, 5, 3, 1, 1, 7, 2, 1, 0, 6, 9, 3, 7, 6, 4, 4, 8, 6, 1, 0
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OFFSET

0,1


COMMENTS

Or, decimal expansion of Pi csch Pi.
Pi csch Pi = Prod[from n = 2 to infinity] (n^2  1)/(n^2 + 1). This is one of an infinite set of infinite products, the next being Prod[from n = 2 to infinity] (n^3  1)/(n^3 + 1) = 2/3. This is related to Ramanujan's surprising formula: Prod[from n = 1 to infinity] (prime(n)^2  1)/(prime(n)^2 + 1) = 5/2 and we use it in finding A112407 the semiprime analog.  Jonathan Vos Post, Dec 07 2005


REFERENCES

Borwein, J.; Bailey, D.; and Girgensohn, R. "Two Products." Section 1.2 in Experimentation in Mathematics: Computational Paths to Discovery. Natick, MA: A. K. Peters, pp. 47, 2004.


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Infinite Product.
Eric Weisstein's World of Mathematics, Hyperbolic Cosecant


FORMULA

Pi/sinh(Pi) = Prod_{k>=1} k^2/(k^2+1) = 0.27202905498213316295...
Equals Gamma(1+i)*Gamma(1i), where i is the imaginary unit.  Vaclav Kotesovec, Dec 10 2015


EXAMPLE

0.272029054982133162950236583672...


MATHEMATICA

Re[N[Gamma[1+I]*Gamma[1I], 104]] (* Vaclav Kotesovec, Dec 09 2015 *)
RealDigits[Pi/Sinh[Pi], 10, 120][[1]] (* Harvey P. Dale, May 16 2019 *)


PROG

(PARI) default(realprecision, 100); Pi/sinh(Pi) \\ G. C. Greubel, Feb 02 2019
(MAGMA) SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)/Sinh(Pi(R)); // G. C. Greubel, Feb 02 2019
(Sage) numerical_approx(pi/sinh(pi), digits=100) # G. C. Greubel, Feb 02 2019


CROSSREFS

Cf. A114528A114536.
Sequence in context: A060465 A219177 A139339 * A245221 A195726 A095194
Adjacent sequences: A090983 A090984 A090985 * A090987 A090988 A090989


KEYWORD

cons,nonn


AUTHOR

Benoit Cloitre, Feb 28 2004


STATUS

approved



