Theorems V in [this paper][1] of [L.E. Dickson][2] states that the following two sets are equal. $$E=\{a^2+b^2+2c^2 \ | \ a,b,c \in \mathbb{Z}\} \  \text{ and } \ F=\mathbb{N} \setminus \{4^k(16n+14) \ | \ k,n \in \mathbb{N}\}.$$
Let $\mathbb{A}$ denote $\mathbb{N}$ or $\mathbb{Z}$. Consider the following set (where $u \cdot v$ denotes the usual [dot product][3]):   $$E(\mathbb{A}) = \{\|u \|^2 + |u \cdot v| \text{ such that } u,v \in \mathbb{A}^3 \text{ and } \|u \| = \|v \|\}.$$

Let $u= (a,b,c)$, $v = (b,-a,c) \in \mathbb{Z}^3$. Then $\|u \| = \|v \|$ and $\|u \|^2 + |u \cdot v| = a^2+b^2+2c^2.$
It follows that $E \subseteq E(\mathbb{Z})$. But surprisingly, the computation below suggests that $E(\mathbb{Z})=F$ also, which would be an extension of Dickson's theorem.   

**Question 1**: Is it true that $E(\mathbb{Z})=F$?
  
The computation suggests also the following question:   

**Question 2**: Is it true that $E(\mathbb{N}) = F \setminus \{ 5, 23, 29, 65, 167 \} $? 

*Application*: [this answer][4] proves that the form $\| A\|^2$ covers every natural number for $A \in M_3(\mathbb{Z})$.  
 A positive answer to Question 2 would prove this result for $A \in M_3(\mathbb{N})$.  

___
**Computation**   


    sage: L=[]
    ....: for a1 in range(-12,12):
    ....:     for a2 in range(-12,12):
    ....:         for a3 in range(-12,12):
    ....:             for b1 in range(-12,12):
    ....:                 for b2 in range(-12,12):
    ....:                     for b3 in range(-12,12):
    ....:                         if a1**2+a2**2+a3**2==b1**2+b2**2+b3**2:
    ....:                             n=a1**2+a2**2+a3**2+a1*b1+a2*b2+a3*b3
    ....:                             L.append(n)
    ....: l=list(set(L))
    ....: l.sort()
    ....: set(range(255))-set(l)
    ....:
    {14, 30, 46, 56, 62, 78, 94, 110, 120, 126, 142, 158, 174, 184, 190, 206, 222, 224, 238, 248, 254}

The research of the above sequence in OEIS provides exactly one result:   
 [A055039][5] `Numbers of the form 2^(2i+1)*(8*j+7)`.

    sage: L=[]
    ....: for a1 in range(25):
    ....:     for a2 in range(25):
    ....:         for a3 in range(25):
    ....:             for b1 in range(25):
    ....:                 for b2 in range(25):
    ....:                     for b3 in range(25):
    ....:                         if a1**2+a2**2+a3**2==b1**2+b2**2+b3**2:
    ....:                             n=a1**2+a2**2+a3**2+a1*b1+a2*b2+a3*b3
    ....:                             L.append(n)
    ....: l=list(set(L))
    ....: l.sort()
    ....: s=set(range(1484))-set(l)
    ....: S=[]
    ....: for i in s:
    ....:     f=list(factor(i))
    ....:     a=f[0][0]
    ....:     b=f[0][1]
    ....:     if a<>2:
    ....:         S.append(i)
    ....:     elif Integer(b).mod(2)==0:
    ....:         S.append(i)
    ....:     elif Integer(i/(2**b)).mod(8)<>7:
    ....:         S.append(i)
    ....: S.sort()
    ....: S
    ....:
    [5, 23, 29, 65, 167]


  [1]: https://mathscinet.ams.org/mathscinet-getitem?mr=1561323
  [2]: https://en.wikipedia.org/wiki/Leonard_Eugene_Dickson
  [3]: https://en.wikipedia.org/wiki/Dot_product
  [4]: https://math.stackexchange.com/a/2967989/84284
  [5]: http://oeis.org/A055039