|\^/|     Maple 2022 (X86 64 LINUX)
._|\|   |/|_. Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 2022
 \  MAPLE  /  All rights reserved. Maple is a trademark of
 <____ ____>  Waterloo Maple Inc.
      |       Type ? for help.

> GetSymMat := proc(n,x)
> local A,i,j,k;
>     A := Matrix(n,n);
>     k := 1;
>     for i to n do
>         A[i,i] := x[k]; k++;
>         for j from i+1 to n do 
>             A[i,j] := x[k];
>             A[j,i] := x[k];
>             k++;
>         od;
>     od;
>     A;
> end:

> numterms := proc(f) 
>    if f=0 then 0 elif type(f,`+`) then nops(f) else 1 fi; 
> end:

> with(LinearAlgebra):

# Test
> S3 := GetSymMat(3,x);
                                [x[1]    x[2]    x[3]]
                                [                    ]
                          S3 := [x[2]    x[4]    x[5]]
                                [                    ]
                                [x[3]    x[5]    x[6]]

> det := Determinant(S3);
                                 2       2                               2
det := x[1] x[4] x[6] - x[1] x[5]  - x[2]  x[6] + 2 x[2] x[3] x[5] - x[3]  x[4]

> numterms(det);
                                       5


> for n from 3 to 9 do
>     Sn := GetSymMat(n,x);
>     det := Determinant(Sn);
>     printf("n=%d  #det(Sn)=%d\n",n,numterms(det));
> od:
n=3  #det(Sn)=5
n=4  #det(Sn)=17
n=5  #det(Sn)=73
n=6  #det(Sn)=388
n=7  #det(Sn)=2461
n=8  #det(Sn)=18155
memory used=300.8MB, alloc=270.2MB, time=2.21
n=9  #det(Sn)=152531
> quit
memory used=458.3MB, alloc=316.9MB, time=3.04