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

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

0,1

Links

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

Formula

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

Example

0.38853615333517585918432957568703590501390...

Maple

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

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A046990, A046991.

Cf. A051762, A085365, A246945, A230821, A249673.

Cf. A027641, A027642.

Sequence in context: A378677 A371941 A232182 * A179553 A157471 A288094

Adjacent sequences: A118814 A118815 A118816 * A118818 A118819 A118820

Keyword

cons,nonn

Author

Fredrik Johansson, May 23 2006

Extensions

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

Status

approved