A269611 Decimal expansion of Sum_{n>=1} (sin(Pi/n))^2.

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, 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, 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

Offset: 1

Author

Vaclav Kotesovec, Mar 01 2016

Keywords

• nonn
• cons

Examples

			4.32267504323963714111855606344042809207852173550531955525699965992300301...
		

Crossrefs

Cf. A051762, A085365, A093721, A269574, A269720.

Programs

Maple

evalf(Sum((sin(Pi/n))^2, n=1..infinity), 120);

Mathematica

RealDigits[NSum[Sin[Pi/n]^2, {n, 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 10000, PrecisionGoal -> 120, Method -> {"NIntegrate", "MaxRecursion" -> 100}]][[1]]

PARI

default(realprecision,120); sumpos(n=1, (sin(Pi/n))^2)

Formula

Equals (1/2) * Sum_{n>=1} (1 - cos(2*Pi/n)).

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.

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).