Let $a,b,c,d,e,f$ be integers with $a\ge c\ge e>0$, $b>-a$ and $a\equiv b\pmod2$, $d>-c$ and $c\equiv d\pmod 2$, $f>-e$ and $e\equiv f\pmod2$. If each $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as $$\frac{x(ax+b)}2+\frac{y(cy+d)}2+\frac{z(ez+f)}2\ \ \ \text{with}\ x,y,z\in\mathbb N,$$ then we say that the ordered tuple $(a,b,c,d,e,f)$ is universal over $\mathbb N$. For example, Gauss' triangular number theorem indicates that $(1,1,1,1,1,1)$ is universal over $\mathbb N$. In a recent paper of mine, I found all candidates for tuples $(a,b,c,d,e,f)$ universal over $\mathbb N$. When $a\mid b$, $c\mid d$ and $e\mid f$, I have shown that many candidates are indeed universal over $\mathbb N$, but the following 10 candidates \begin{align}&(4,0,2,0,1,3),\, (4,0,2,0,1,5),\,(4,0,2,6,1,1),\, (4,0,2,6,2,0), \,(4,4,2,0,1,3), \\&(4,8,2,0,1,1),\, (4,8,2,0,1,3), \,(4,12,2,0,1,1),\, (6,0,2,0,1,3), \,(6,6,2,0,1,3) \end{align} have not yet been proved to be universal over $\mathbb N$.
Conjecture. All the 10 tuples listed above are universal over $\mathbb N$.
Note that $(4,8,2,0,1,3)$ is universal over $\mathbb N$ if and only if any integer $n\ge3$ can be written as $x^2+2y^2+z(z+1)/2$ with $x\in\mathbb N$ and $y,z\in\mathbb Z^+=\{1,2,3,\ldots\}$.
Any ideas to solve the conjecture?
When $a\nmid b$ or $c\nmid d$ or $e\nmid f$, I have shown in the same paper that there are 407 candidates for such tuples $(a,b,c,d,e,f)$ universal over $\mathbb N$. It seems that none of the 407 tuples (listed in the Appendix of my paper) can be easily proved to be universal over $\mathbb N$.
