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A118817
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Decimal expansion of Product_{n >= 1} cos(1/n).
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5
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3, 8, 8, 5, 3, 6, 1, 5, 3, 3, 3, 5, 1, 7, 5, 8, 5, 9, 1, 8, 4, 3, 2, 9, 5, 7, 5, 6, 8, 7, 0, 3, 5, 9, 0, 5, 0, 1, 3, 9, 0, 0, 5, 2, 8, 5, 9, 7, 5, 1, 7, 9, 2, 1, 9, 1, 3, 1, 8, 4, 6, 1, 1, 9, 9, 8, 7, 9, 8, 7, 4, 9, 4, 3, 4, 6, 3, 3, 9, 3, 2, 7, 6, 8, 3, 8, 8, 4, 3, 1, 9, 7, 8, 1, 3, 8, 3, 4, 0, 8, 2, 2, 4, 1, 3
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OFFSET
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0,1
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LINKS
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FORMULA
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Equals exp(Sum_{n>=1} (-c(n)*zeta(2n)), where c(n) = A046990(n)/A046991(n).
Equals exp(-Sum_{n>=1} (2^(2*n)-1) * Zeta(2*n)^2 / (n*Pi^(2*n)) ). - Vaclav Kotesovec, Sep 20 2014
Equals exp(Sum_{k>=1} (-1)^k*2^(2*k-1)*(2^(2*k)-1)*B(2*k)*zeta(2*k)/(k*(2*k)!)), where B(k) is the k-th Bernoulli number. - Amiram Eldar, Jul 30 2023
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EXAMPLE
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0.38853615333517585918432957568703590501390...
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MAPLE
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nn:= 120:
p:= product(cos(1/n), n=1..infinity):
f:= evalf(p, nn+10):
s:= convert(f, string):
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MATHEMATICA
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S = Series[Log[Cos[x]], {x, 0, 400}]; N[Exp[N[Sum[SeriesCoefficient[S, 2k] Zeta[2k], {k, 1, 200}], 70]], 50]
Block[{$MaxExtraPrecision = 1000}, Do[Print[N[1/Exp[Sum[(2^(2*n) - 1)*Zeta[2*n]^2/(n*Pi^(2*n)), {n, 1, m}]], 110]], {m, 250, 300}]] (* Vaclav Kotesovec, Sep 20 2014 *)
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PROG
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(PARI) exp(-sumpos(n=1, -log(cos(1/n)))) \\ warning: requires 2.6.2 or greater; Charles R Greathouse IV, Nov 04 2013
(PARI) T(n)=((-4)^n-(-16)^n)*bernfrac(2*n)/2/n/(2*n)!
lm=lambertw(2*log(Pi/2)*10^default(realprecision))/2/log(Pi/2); exp(-sum(n=1, lm, T(n)*zeta(2*n))) \\ Charles R Greathouse IV, Nov 06 2013
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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