A067754 Number of unordered primitive solutions to xy+xz+yz=n.

1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 4, 2, 3, 2, 3, 3, 4, 2, 4, 4, 2, 4, 3, 2, 4, 4, 4, 3, 4, 3, 6, 3, 2, 4, 6, 4, 5, 4, 3, 4, 4, 3, 6, 4, 3, 4, 6, 3, 4, 4, 6, 6, 4, 2, 7, 4, 4, 5, 6, 3, 6, 6, 3, 5, 6, 4, 8, 4, 3, 6, 6, 4, 6, 4, 6, 6, 4, 3, 7, 6, 4, 6, 8, 4, 7, 6, 6, 4, 4, 5, 10, 6, 3, 5, 6, 3

Offset: 1

Author

Colin Mallows, Jan 31 2002

Keywords

• easy
• nonn

Comments

For n = m^2 this is the number of root Descartes quadruples (-m,b,c,d).

An upper bound on the number of solutions appears to be 1.5*sqrt(n). - T. D. Noe, Jun 14 2006

Examples

			a(9)=2 because of (0,1,9) and (1,1,4) (but not (0,3,3)).
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks, and C. H. Yan, Apollonian Circle Packings: Number Theory, J. Number Theory, 100 (2003), 1-45.

Crossrefs

Cf. A067751, A067752, A067753.

Programs

Mathematica

Table[cnt=0; Do[z=(n-x*y)/(x+y); If[IntegerQ[z] && GCD[x,y,z]==1, cnt++ ], {x,0,Sqrt[n/3]}, {y, Max[1,x],Sqrt[x^2+n]-x}]; cnt, {n,100}] (* T. D. Noe, Jun 14 2006 *)

Formula

a(n) = A066360(n) + A007875(n). - T. D. Noe, Jun 14 2006

Extensions

Corrected and extended by T. D. Noe, Jun 14 2006