Restriction of Jacobi's four-squares theorem

Question

We know that the number of decomposition as a sum of four squares of $n\in\mathbb{N}$ such that $n=a^2 + b^2 + c^2 + d^2$ is : $$ r_4(n) = 8 \sum_{d\mid n, 4\nmid d}{d} $$ And there is a more general one from this answer.

But is there any restriction of this function to $a,b,c,d\in\mathbb{N}^*$ ?

Answer 1

By inclusion-exclusion principle, the number of representations of $n$ as the sum of squares of four nonzero integers equals: $$\sum_{k=0}^4 \binom4k (-1)^k r_{4-k}(n).$$ Formulae for $r_k(n)$ are given in this article at MathWorld.

If one wants to further restrict the representations to positive integers, the above expression needs to be divided by $2^4=16$. Numerical values are given in A063730.