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Consider a quadratic form $Q(a,b,c,d)=ab+ac+ad+bc+bd+cd$ on $\mathbb Z^4$.

For some reason I am interested in the number of solutions $(a,b,c,d)\in\mathbb Z_{> 0}^4$ of $Q(a,b,c,d)=n$ as a function of $n\in \mathbb Z_{> 0}$.

Is there a closed formula (may be not explicit, e.g. through some theta functions?) Is there a way to obtain an asymptotic?

Improve this question
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    $\begingroup$ Not sure if it helps, but this form coincides with $\frac12((a+b+c+d)^2-a^2-b^2-c^2-d^2)$. $\endgroup$
    – Wojowu
    Aug 21, 2021 at 15:50
  • $\begingroup$ @Wojowu I would guess that this form is useless since $a+b+c+d,a,b,c,d$ are not linearly independent. Think of the "$n=2$" case: it does not help to rewrite $ab=n$ as $(a+b)^2-a^2-b^2=2n$. $\endgroup$
    – Z. M
    Aug 21, 2021 at 16:39
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    $\begingroup$ The first hundert terms for $n=1,\ldots,100$ are: $6, 12, 16, 18, 24, 25, 24, 36, 34, 24, 48, 44, 30, 48, 52, 42, 60, 44, 48, 72, 64, 30, 84, 77, 42, 72, 80, 60, 96, 56, 72, 96, 82, 48, 120, 102, 54, 84, 112, 72, 132, 80, 72, 132, 104, 54, 156, 108, 90, 96, 136, 90, 144, 105, 96, 168, 112, 48, 192, 132, 102, 120, 176, 102, 168, 104, 96, 180, 166, 72, 228, 148, 102, 144, 152, 132, 216, 110, 144, 192, 182, 72, 240, 192, 96, 156, 220, 102, 252, 136, 168, 216, 160, 102, 264, 213, 126, 180, 224, 126$ (This sequence is not recognized by the OEIS!) $\endgroup$
    – Roland Bacher
    Aug 21, 2021 at 20:51
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    $\begingroup$ The title asks for positive integer solutions, but the body allows solutions involving zero. In some parts of the mathematical world, zero is not considered to be a positive integer. $\endgroup$
    – Gerry Myerson
    Aug 21, 2021 at 23:24
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    $\begingroup$ @Will, yes, oeis.org/A025052 is a better reference, as it gives links to several papers about $ab+ac+bc=n$. $\endgroup$
    – Gerry Myerson
    Aug 22, 2021 at 3:10

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