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A227002
Number of integer solutions to n = f(x,y,z) = x*(x-y)*(x-z) + y*(y-z)*(y-x) + z*(z-x)*(z-y) with 0 <= x <= y <= z.
1
2, 2, 3, 3, 3, 3, 3, 4, 5, 2, 3, 6, 3, 2, 4, 6, 3, 4, 4, 5, 4, 2, 4, 6, 4, 3, 6, 4, 3, 5, 3, 6, 6, 2, 3, 9, 4, 2, 4, 6, 3, 4, 3, 5, 7, 3, 4, 8, 4, 3, 7, 4, 3, 6, 3, 6, 4, 4, 4, 6, 4, 2, 7, 8, 4, 3, 3, 6, 4, 3, 4, 14, 4, 2, 6, 4, 3, 3, 4, 7, 8, 4, 3, 7, 4, 3, 5
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
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
Sequence in context: A262685 A171895 A025785 * A343118 A087182 A035453
KEYWORD
nonn
AUTHOR
Michael Somos, Jun 26 2013
STATUS
approved