A230821 -id:A230821 - cp's OEIS frontend

A249673 Decimal expansion of Product_{n>=1} cosh(1/n).

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

Offset: 1

Vaclav Kotesovec, Nov 03 2014

			2.116465536505484775878572227025831988148089392809082568281348...

Cf. A118817, A230821, A051762, A218504, A246945, A270614.

Cf. A027641, A027642.

evalf(exp(sum(log(cosh(1/n)), n=1..infinity)), 100)

default(realprecision,120); exp(sumpos(k=1, log(cosh(1/k))))

From Amiram Eldar, Jul 30 2023: (Start)
Equals exp(Sum_{k>=1} 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.
Equals exp(Sum_{k>=1} (-1)^(k+1)*(2^(2*k)-1)*zeta(2*k)^2/(k*Pi^(2*k))). (End)

A118817 Decimal expansion of Product_{n >= 1} cos(1/n).

Original entry on oeis.org

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

Offset: 0

Fredrik Johansson, May 23 2006

			0.38853615333517585918432957568703590501390...

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A046990, A046991.
Cf. A051762, A085365, A246945, A230821, A249673.
Cf. A027641, A027642.

nn:= 120:
p:= product(cos(1/n), n=1..infinity):
f:= evalf(p, nn+10):
s:= convert(f, string):
seq(parse(s[n+1]), n=1..nn);  # Alois P. Heinz, Nov 04 2013

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 *)

exp(-sumpos(n=1,-log(cos(1/n)))) \\ warning: requires 2.6.2 or greater; Charles R Greathouse IV, Nov 04 2013

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

Equals exp(Sum_{n>=1} -c(n)*zeta(2*n)), 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

Corrected offset and extended by Robert G. Wilson v, Nov 03 2013

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A249673 Decimal expansion of Product_{n>=1} cosh(1/n).

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A118817 Decimal expansion of Product_{n >= 1} cos(1/n).

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