A086215 Number of (-1,0,1) n X n matrices M that are positive definite.

1, 7, 311, 79505, 105311665, 642005451319, 15477341239385927

Offset

1,2

Comments

M need not be symmetric. For the number of different values of M + M' see A114601. - Max Alekseyev, Dec 13 2005

Links

Table of n, a(n) for n=1..7.

Eric Weisstein's World of Mathematics, Positive Definite Matrix

Mathematica

Table[Count[Tuples[{-1, 0, 1}, {n, n}], _?PositiveDefiniteMatrixQ], {n, 3}] (* Eric W. Weisstein, Jan 03 2021 *)

Prog

(PARI) { a(n) = M=matrix(n, n, i, j, 2*(i==j)); r=0; b(1); r } { b(k) = local(z, t); if(k>n, z=t=0; for(i=1, n, for(j=1, i-1, if(M[ i, j ]==0, z++); if(abs(M[ i, j ])==1, t++); )); r+=3^z*2^t; return; ); forvec(x=vector(k-1, i, [ -1, 1 ]), for(i=1, k-1, M[ k, i ]=M[ i, k ]=x[ i ]); if( matdet(vecextract(M, 2^k-1, 2^k-1), 1)>0, b(k+1) ) ) } /* Max Alekseyev */

Crossrefs

Cf. A114601, A085656.

Sequence in context: A002437 A300870 A239163 * A119163 A364479 A171148

Adjacent sequences: A086212 A086213 A086214 * A086216 A086217 A086218

Keyword

nonn,hard,more

Author

Eric W. Weisstein, Jul 12 2003

Extensions

a(4) from Wouter Meeussen, Sep 05 2003

a(5)-a(6) from Max Alekseyev, Dec 13 2005

a(7) from Max Alekseyev, Nov 09 2006

Status

approved