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%I #13 Mar 27 2024 20:11:13
%S 4,3,2,2,6,7,5,0,4,3,2,3,9,6,3,7,1,4,1,1,1,8,5,5,6,0,6,3,4,4,0,4,2,8,
%T 0,9,2,0,7,8,5,2,1,7,3,5,5,0,5,3,1,9,5,5,5,2,5,6,9,9,9,6,5,9,9,2,3,0,
%U 0,3,0,1,0,6,1,4,8,2,3,0,7,9,8,4,1,1,0,7,7,0,5,8,5,1,5,0,2,6,3,5,0,8,1,4,7
%N Decimal expansion of Sum_{n>=1} (sin(Pi/n))^2.
%F Equals (1/2) * Sum_{n>=1} (1 - cos(2*Pi/n)).
%F Equals Sum_{k>=1} (-1)^(k+1) * 2^(2*k-1) * Pi^(2*k) * Zeta(2*k) / (2*k)!, where Zeta is the Riemann zeta function.
%F Equals Sum_{k>=1} 2^(4*k-2) * Pi^(4*k) * B(2*k) / (2*k)!^2, where B(n) is the Bernoulli number A027641(n)/A027642(n).
%e 4.32267504323963714111855606344042809207852173550531955525699965992300301...
%p evalf(Sum((sin(Pi/n))^2, n=1..infinity), 120);
%t RealDigits[NSum[Sin[Pi/n]^2, {n, 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 10000, PrecisionGoal -> 120, Method -> {"NIntegrate", "MaxRecursion" -> 100}]][[1]]
%o (PARI) default(realprecision,120); sumpos(n=1, (sin(Pi/n))^2)
%Y Cf. A051762, A085365, A093721, A269574, A269720.
%K nonn,cons
%O 1,1
%A _Vaclav Kotesovec_, Mar 01 2016
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