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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 (list; graph; refs; listen; history; text; internal format)
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 non-primitive 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=(n-x*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

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Last modified January 19 09:35 EST 2020. Contains 331048 sequences. (Running on oeis4.)