

A067752


Number of unordered solutions of xy + xz + yz = n in nonnegative integers.


5



1, 1, 2, 2, 2, 2, 2, 3, 3, 2, 3, 4, 2, 3, 4, 4, 3, 3, 3, 5, 4, 2, 4, 6, 3, 4, 5, 4, 4, 4, 4, 6, 4, 3, 6, 7, 2, 4, 6, 6, 5, 4, 3, 7, 6, 3, 6, 8, 4, 5, 6, 5, 4, 6, 6, 9, 4, 2, 7, 8, 4, 5, 8, 7, 6, 6, 3, 8, 6, 4, 8, 9, 3, 6, 8, 7, 6, 4, 6, 11, 7, 3, 7, 10, 4, 6, 8, 6, 7
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OFFSET

1,3


COMMENTS

An upper bound on the number of solutions appears to be 1.5*sqrt(n).  T. D. Noe, Jun 14 2006
a(n) is also the total number of distinct quadratic forms of discriminant 4n. A232551 counts only the primitive quadratic forms of discriminant 4n (those with all coefficients pairwise coprime) and A234287 includes those by which some prime can be represented (those with all coefficients pairwise coprime or coefficient of x^2 is prime or coefficient of y^2 is prime). This sequence includes all quadratic forms like 2x^2 + 2xy + 4y^2 and 2x^2 + 4y^2 which are nonprimitive and those like 4x^2 + 2xy + 4y^2 and 4x^2 + 4xy + 4y^2 by which no prime can be represented (those with no restrictions).  V. Raman, Dec 24 2013


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000
V. Raman, Examples of these distinct quadratic forms for n = 1..100


EXAMPLE

a(12)=4 because of (0,1,12), (0,2,6), (0,3,4), (2,2,2).
a(20)=5 because of (0,1,20), (0,2,10), (0,4,5), (1,2,6), (2,2,4).


MATHEMATICA

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


CROSSREFS

Cf. A000003, A000926, A067751, A067753, A067754, A232550, A232551, A234287.
Sequence in context: A320111 A234287 A084294 * A229942 A025422 A078640
Adjacent sequences: A067749 A067750 A067751 * A067753 A067754 A067755


KEYWORD

easy,nonn


AUTHOR

Colin Mallows, Jan 31 2002


EXTENSIONS

Corrected, extended and edited by John W. Layman, Dec 03 2004


STATUS

approved



