OFFSET
1,3
LINKS
E. Deutsch and M. S. Klamkin, Counting the Solutions: Problem 10540, Amer. Math. Monthly 108, (2001), p. 172.
FORMULA
a(n)=denominator of [n(n-1)(n^2-n-1)+4sum(floor(n/(k^2))*phi(k), k=2...floor(sqrt(n)))-2sum(floor(n/k)^2*phi(k), k=2... n)]/(2n^4).
EXAMPLE
a(3)=27 because at the 81 quadruples (x,y,z,u) (1<=x,y,z,u<=3) the function
(x-y)/(x+y)+(y-z)/(y+z)+(z-u)/(z+u)+(u-x)/(u+x) assumes twelve times the value 1/30, twelve times the value -1/30 and fifty-seven times the value 0; then the considered probability is 12/81=4/27.
0,0,4/27,15/64,36/125,103/324,832/2401
MAPLE
with(numtheory): a:=n*(n-1)*(n^2-n-1): b:=4*sum(floor(n/k^2)*phi(k), k=2..floor(sqrt(n))): c:=2*sum((floor(n/k))^2*phi(k), k=2..n): p:=proc(n) (a+b-c)/2/n^4 end: seq(denom(simplify(p(n))), n=1..45);
CROSSREFS
KEYWORD
frac,nonn
AUTHOR
Emeric Deutsch, Apr 24 2005
STATUS
approved