A180307 Decimal expansion of the mean length of a line segment picked at random in a 3, 4, 5 (right) triangle.

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

Offset: 1

Author

Eric W. Weisstein, Aug 25 2010

Keywords

• nonn
• cons

Examples

			1.4581846347360227433...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, 3, 4, 5 Triangle
• Eric Weisstein's World of Mathematics, Triangle Line Picking

Crossrefs

Cf. A093063.

Programs

Magma

SetDefaultRealField(RealField(111)); (20460 +9728*Log(2) +5103*Log(3) )/22500; // G. C. Greubel, Dec 20 2019

Maple

evalf( (20460+9728*log(2)+5103*log(3))/22500, 111); # G. C. Greubel, Dec 20 2019

Mathematica

F[a_, b_, c_]:= (a^3 +b^3 +2*c^3)/(15*c^2) +(a^2/(15*b))*(1 + (b/c)^3)* ArcCsch[a/b] +(b^2/(15*a))*(1 +(a/c)^3)*ArcCsch[b/a]; RealDigits[F[3, 4, 5], 10, 110][[1]] (* G. C. Greubel, Dec 20 2019 *)

PARI

arcsch(z)=log(1/z+sqrt(1/z^2+1));
seglen(a,b)={my(c=sqrt(a^2+b^2));(a^3+b^3+2*c^3)/(15*c^2)+(a^2/(15*b))*(1+(b/c)^3)*arcsch(a/b)+(b^2/(15*a))*(1+(a/c)^3)*arcsch(b/a)};
seglen(3,4) \\ Hugo Pfoertner, Dec 18 2019

Sage

def F(a, b, c): return (a^3 + b^3 + 2*c^3)/(15*c^2) + (a^2/(15*b))*(1 + (b/c)^3)*arccsch(a/b) + (b^2/(15*a))*(1 + (a/c)^3)*arccsch(b/a)
numerical_approx(F(3,4,5), digits=110) # G. C. Greubel, Dec 20 2019

Formula

Equals (20460 + 9728*log(2) + 5103*log(3))/22500.

Equals (a^3 + b^3 + 2*c^3) / (15*c^2) + (a^2 / (15*b)) * (1 + (b/c)^3) * cosech^{-1}(a/b) + (b^2 / (15*a)) * (1 + (a/c)^3) * cosech^{-1}(b/a) for an arbitrary right angled triangle with sides a, b and (hypotenuse) c. - Muthu Veerappan Ramalingam, Dec 18 2019