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A067752 Number of unordered solutions of xy + xz + yz = n in nonnegative integers. 5

%I #29 Jul 25 2017 02:47:31

%S 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,

%T 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,

%U 6,4,8,9,3,6,8,7,6,4,6,11,7,3,7,10,4,6,8,6,7

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

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

%C 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

%H T. D. Noe, <a href="/A067752/b067752.txt">Table of n, a(n) for n = 1..10000</a>

%H V. Raman, <a href="/A067752/a067752.txt">Examples of these distinct quadratic forms for n = 1..100</a>

%e a(12)=4 because of (0,1,12), (0,2,6), (0,3,4), (2,2,2).

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

%t 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 *)

%Y Cf. A000003, A000926, A067751, A067753, A067754, A232550, A232551, A234287.

%K easy,nonn

%O 1,3

%A _Colin Mallows_, Jan 31 2002

%E Corrected, extended and edited by _John W. Layman_, Dec 03 2004

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