Symmetric Numerical Semigroups for a fixed Embedding Dimension

Question

I was trying to find some symmetric Numerical Semigroups with embedding dimensions $5,6,7,...$ so on, like $S:=\langle 6,17,27,28,38\rangle$ (Frobenius Number = $49$).

Now, I want to know if there is any general formula to find explicitly the list of the all possible symmetric Numerical Semigroups for a fixed embedding dimension or any general form.

I got a general form of symmetric Numerical Semigroups in the book "Numerical Semigroups by J.C. Rosales & P.A. Garcia Sanchez". The form is like this,

$S:=\langle m,m+1,qm+2q+2, . . . ,qm+(m−1)\rangle$ ;
Multiplicity '$m$', embedding dimension '$(m-2q)$', Frobenius Number '$(2qm+2q+1)$'.

But, I want all the symmetric Numerical Semigroups or any general form for a fixed Embedding Dimension otherthan this.

Can anyone help me ?

Thanks in advance.

Answer 1

I think another general form of such symmetric numerical semigroup can also be easily obtained using some Arithmatic Progression. Like, the Numerical Semigroup :-

$S:=\langle m,(m+(2\times1)),(m+(2\times2)),....,(m+2\times(m-2))\rangle$ (where, $m$ is odd). Check $S$ is symmetric.

Also you can simply manipulate the form that you have given to get some new types.

Otherwise, I think there is no such rules to get those general forms easily.