minimum of 4 terms (A086215= 1, 7, 311) (longish)

[wouter meeussen]

is a wise rule.
So I tried to extend it.

%I A086215
%S A086215 1, 7, 311
%N A086215 Number of (-1, 0, 1) n X n matrices that are positive definite.
%H A086215 Eric Weisstein's World of Mathematics, < a href =
  "http://mathworld.wolfram.com/PositiveDefiniteMatrix.html" >
    Positive Definite Matrix < /a >
%Y A086215 Sequence in context :
        A082168 A015005 A002437 this_sequence A082160 A002000 A049686
%Y A086215 Adjacent sequences :
    A086212 A086213 A086214 this_sequence A086216 A086217 A086218
%K A086215 nonn, more
%O A086215 1, 2
%A A086215 Eric W. Weisstein (eric(AT)weisstein.com), Jul 12, 2003

and found a(4)=79505 after a bit over an hour.

But ... it needs checking, since the listing of the eigenvalues depends on numeric, not symbolic
computation. These eigenvalues then need to be checked for being real and strictly positive, and an
error close to zero is not uncommon in such calculations.

Therefore I tried a different attack :
instead of
generating all 729*729 (-1,0,1)'P' matrices by adding all 729 lower triangular ones to their
transposes in all possible ways, and then adding to that all 31 diagonal (-1,0,1)'D' matrices with
positive trace (pigeonhole:: all eigenvalues positive implies positive trace), and then count those
with all eigenvalues >0 , or equivalently,
generate T= P+D; count the T that satisfy Eigenvalues of (T+transpose(T))/2 >0
I rather did:
first take the union of all  P+transpose(P) -matrices, and combine those with the diagonal
D-matrices into T'-matrices, only then count the T' that satisfy Eigenvalues of T' >0.

What's so different?
Aha, there are only 393 different T' matrices we need to calculate eigenvalues for.
but each of those can be generated by many different P-matrices.

8 T-matrices, each resulting from 64 different P matrices giving the same (P+tr_P) =512
48 T-matrices, each from 96 different P matrices =4608
120 T-matrices, each from 144 different P matrices =17280
144 T-matrices, each from 216 different P matrices =31104
60 T-matrices, each from 324 different P matrices =19440
12 T-matrices, each from 486 different P matrices =5832
1 T-matrix, resulting from 729 different P matrices giving the same (P+tr_P) =729

8+48+120+144+60+12+1 =393, and
512+4608+17280+31104+19440+5832+729 =79505 (* so that checks out*)
the lowest eigenvalue(s) are 0.133975, well away fom zero, so that checks out too.

ergo : %S A086215 1, 7, 311, 79505

Wouter.