cp's OEIS Frontend

A144663 Decimal expansion of Product_{n>=2} (n^4-1)/(n^4+1).

%I A144663 #24 Feb 16 2025 08:33:09
%S A144663 8,4,8,0,5,4,0,4,9,3,5,2,9,0,0,3,9,2,1,2,9,6,5,0,1,8,3,4,0,5,0,0,7,7,
%T A144663 0,5,8,4,7,9,8,7,4,8,6,8,8,4,7,1,7,6,6,6,4,3,0,6,9,6,4,5,3,8,0,6,6,1,
%U A144663 3,5,7,2,8,5,5,5,5,4,4,1,2,7,1,3,6,7,6,6,3,7,6,7,3,6,9,0,1,2,5,2,9,5,8,7,6
%N A144663 Decimal expansion of Product_{n>=2} (n^4-1)/(n^4+1).
%H A144663 J. Borwein et al., <a href="http://www.ams.org/mathscinet-getitem?mr=2051473">Experimentation in Mathematics</a>, 2004, section 1.2.
%H A144663 László Tóth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Toth/toth46.html">Transcendental Infinite Products Associated with the ±1 Thue-Morse Sequence</a>, Journal of Integer Sequences, Vol. 23 (2020), Article 20.8.2.
%H A144663 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>.
%F A144663 Equals Pi*sinh(Pi) / (cosh(sqrt(2)*Pi) - cos(sqrt(2)*Pi)). - _Vaclav Kotesovec_, Dec 08 2015
%e A144663 0.8480540493529003921296501834...
%p A144663 Digits := 120 :
%p A144663 m := 1:
%p A144663 for r from 2 to 10 do
%p A144663 omega := cos(Pi/r)+I*sin(Pi/r) :
%p A144663 x := (-1)^(m+1)*2*m*m!/r*mul( GAMMA(-m*omega^j)^(-(-1)^j),j=1..2*r-1) ;
%p A144663 x := Re(evalf(x)) ;
%p A144663 print(r,x) ;
%p A144663 od:
%t A144663 RealDigits[ -1/2*Pi*Csc[(-1)^(1/4)*Pi]*Csc[(-1)^(3/4)*Pi]*Sinh[Pi] // Re, 10, 105] // First (* _Jean-François Alcover_, Feb 11 2013 *)
%t A144663 RealDigits[Re[N[Product[(n^4 - 1)/(n^4 + 1), {n, 2, Infinity}], 110]]][[1]] (* _Bruno Berselli_, Apr 02 2013 *)
%o A144663 (PARI) Pi*sinh(Pi)/(cosh(Pi*sqrt(2))-cos(Pi*sqrt(2))) \\ _Michel Marcus_, Sep 07 2020
%Y A144663 Cf. A090986, A255434.
%K A144663 nonn,cons
%O A144663 0,1
%A A144663 _R. J. Mathar_, Feb 01 2009
%E A144663 More terms from _Jean-François Alcover_, Feb 11 2013