OFFSET
1,1
COMMENTS
In Bullen (1998) page 229, the r=1 case of Schur's inequality implies that f(x, y, z) is positive if x, y, z are positive.
n = f(n-1, n-1, n) = f(n, n+1, n+1) so a(n) >= 2 for all n > 0.
0 = f(x, x, x) so there are an infinite number of solutions for n=0.
Except for (x, y, z) = (n, n+1, n+1) all other solutions have z <= n.
f(x, y, z) = x * y * z - (x + y - z) * (y + z - x) * (z + x - y). See Bullen (1998) page 17 Padoa's inequality.
REFERENCES
P. S. Bullen, A dictionary of inequalities, Addison Wesley Longman Limited, 1998.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..200
PROG
(PARI) {a(n) = if( n<1, 0, 1 + sum( z=0, n, sum( y=0, z, sum( x=0, y, n == x*(x - y)*(x - z) + y*(y - z)*(y - x) + z*(z - x)*(z - y)))))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jun 26 2013
STATUS
approved