A118817 - OEIS

%I #38 Sep 13 2024 03:33:12

%S 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,

%T 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,

%U 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

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

%H Charles R Greathouse IV, <a href="/A118817/b118817.txt">Table of n, a(n) for n = 0..10000</a>

%F Equals exp(Sum_{n>=1} -c(n)*zeta(2*n)), where c(n) = A046990(n)/A046991(n).

%F Equals exp(-Sum_{n>=1} (2^(2*n)-1) * Zeta(2*n)^2 / (n*Pi^(2*n)) ). - _Vaclav Kotesovec_, Sep 20 2014

%F 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

%e 0.38853615333517585918432957568703590501390...

%p nn:= 120:

%p p:= product(cos(1/n), n=1..infinity):

%p f:= evalf(p, nn+10):

%p s:= convert(f, string):

%p seq(parse(s[n+1]), n=1..nn);  # _Alois P. Heinz_, Nov 04 2013

%t S = Series[Log[Cos[x]], {x, 0, 400}]; N[Exp[N[Sum[SeriesCoefficient[S, 2k] Zeta[2k], {k, 1, 200}], 70]], 50]

%t 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 *)

%o (PARI) exp(-sumpos(n=1,-log(cos(1/n)))) \\ warning: requires 2.6.2 or greater; _Charles R Greathouse IV_, Nov 04 2013

%o (PARI) T(n)=((-4)^n-(-16)^n)*bernfrac(2*n)/2/n/(2*n)!

%o 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

%Y Cf. A046990, A046991.

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

%Y Cf. A027641, A027642.

%K cons,nonn

%O 0,1

%A _Fredrik Johansson_, May 23 2006

%E Corrected offset and extended by _Robert G. Wilson v_, Nov 03 2013