cp's OEIS Frontend

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

%I A180307 #19 Feb 16 2025 08:33:13
%S A180307 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,
%T A180307 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,
%U A180307 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
%N A180307 Decimal expansion of the mean length of a line segment picked at random in a 3, 4, 5 (right) triangle.
%H A180307 G. C. Greubel, <a href="/A180307/b180307.txt">Table of n, a(n) for n = 1..10000</a>
%H A180307 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/345Triangle.html">3, 4, 5 Triangle</a>
%H A180307 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangleLinePicking.html">Triangle Line Picking</a>
%F A180307 Equals (20460 + 9728*log(2) + 5103*log(3))/22500.
%F A180307 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
%e A180307 1.4581846347360227433...
%p A180307 evalf( (20460+9728*log(2)+5103*log(3))/22500, 111); # _G. C. Greubel_, Dec 20 2019
%t A180307 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 *)
%o A180307 (PARI) arcsch(z)=log(1/z+sqrt(1/z^2+1));
%o A180307 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)};
%o A180307 seglen(3,4) \\ _Hugo Pfoertner_, Dec 18 2019
%o A180307 (Magma) SetDefaultRealField(RealField(111)); (20460 +9728*Log(2) +5103*Log(3) )/22500; // _G. C. Greubel_, Dec 20 2019
%o A180307 (Sage)
%o A180307 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)
%o A180307 numerical_approx(F(3,4,5), digits=110) # _G. C. Greubel_, Dec 20 2019
%Y A180307 Cf. A093063.
%K A180307 nonn,cons
%O A180307 1,2
%A A180307 _Eric W. Weisstein_, Aug 25 2010