A275393 Decimal expansion of the constrained expectation of the product of an angle and the opposite side in a random spherical triangle.

3, 0, 5, 3, 8, 3, 1, 9, 1, 6, 4, 3, 8, 0, 2, 7, 0, 2, 0, 2, 5, 0, 5, 5, 7, 7, 7, 7, 3, 8, 7, 3, 3, 3, 9, 7, 5, 5, 2, 4, 7, 0, 7, 8, 8, 1, 0, 9, 7, 0, 7, 5, 8, 2, 4, 9, 5, 4, 9, 7, 2, 3, 0, 6, 2, 0, 9, 7, 2, 6, 8, 6, 5, 9, 9, 3, 6, 5, 7, 3, 2, 2, 5, 0, 5, 5, 5, 1, 3, 6, 6, 4, 7, 0, 5, 7, 2, 6, 2, 1

Offset

1,1

Comments

Let 'alpha' be an arbitrary angle in a random spherical triangle T and 'a' be the side opposite alpha. (The sphere has radius 1; vertices of T are independent and uniform.) If some other side is constrained to be Pi/2, then E(alpha a) = 3.05... . - [Comment adapted from Steven Finch's abstract]

Links

Table of n, a(n) for n=1..100.

Steven R. Finch, Correlation between Angle and Side , arXiv:1012.0781 [math.PR] 2010.

Formula

(1/4) * integral_{0..Pi} (2- 2F1(1/2,1/2,2,cos(t)^2) cos(t))t dt, where 2F1 is the hypergeometric function.

A faster formula given by David Broadhurst:

- integral_{Pi/2..Pi} (sin(t)+t cos(t))/AGM(1,sin(t)) dt, where AGM is the arithmetic-geometric mean.

Example

3.0538319164380270202505577773873339755247078810970758249549723062...

Mathematica

m = - NIntegrate[(Sin[t] + t Cos[t])/ArithmeticGeometricMean[1, Sin[t]], {t, Pi/2, Pi}, WorkingPrecision -> 100];
RealDigits[m][[1]]

Crossrefs

Sequence in context: A002123 A276408 A225744 * A029840 A144670 A370549

Adjacent sequences: A275390 A275391 A275392 * A275394 A275395 A275396

Keyword

nonn,cons

Author

Jean-François Alcover, Jul 26 2016

Status

approved