Frobenius formula for shifted square sequences beyond 30

Let $a$ be an integer with $a>30$, and let

$$A(a)=(a,a+1^2,a+2^2,a+3^2,\ldots).$$

Let $S(A(a))$ be the numerical semigroup generated by $A(a)$. For $n\in\mathbb{Z}^{+}$, let $\iota(n)$ be the least number of positive integral squares whose sum is $n$. Frobenius formula for shifted square sequences. For every integer $a>30$, the hypotheses of the preceding theorem are satisfied, and consequently

$$F(S(A(a)))=3a+\max\{r\mid 1\leq r\leq a-1,\ \iota(r)=4,\ \iota(a+r)\geq 3,\ \iota(2a+r)\geq 2\}.$$

This gives an explicit formula for the Frobenius number of the numerical semigroup generated by the shifted square sequence when the initial parameter exceeds $30$; the supplied text does not indicate whether this assertion has been independently proved or remains open.