OFFSET
1,2
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
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
EXAMPLE
1.4581846347360227433...
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 *)
PROG
(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
(Magma) SetDefaultRealField(RealField(111)); (20460 +9728*Log(2) +5103*Log(3) )/22500; // G. C. Greubel, Dec 20 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
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Aug 25 2010
STATUS
approved