A114601 -id:A114601 - cp's OEIS frontend

A085657 Number of n X n symmetric positive definite matrices with 2's on the main diagonal and 1's and 0's elsewhere.

1, 2, 8, 61, 819, 17417, 506609, 15582436

Offset: 1

N. J. A. Sloane, Jul 12 2003

nonn

Of course the total number of symmetric matrices of this type (not necessarily positive definite) is 2^C(n,2).

This gives the number of different values of A + A' where A runs through the matrices counted in A085656. - Max Alekseyev, Dec 13 2005

			The singular matrix
2 0 1 1
0 2 1 1
1 1 2 0
1 1 0 2
is one of the three 4 X 4 matrices which are not positive definite.

Index entries for sequences related to binary matrices

Cf. A085659, A080858, A083029.

Cf. A085656, A114601.

{ a(n) = M=matrix(n,n,i,j,2*(i==j)); r=0; b(1); r } { b(k) = if(k>n, r++; return); forvec(x=vector(k-1,i,[0,1]), 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)

More terms from Max Alekseyev, Dec 13 2005

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

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jul 12 2003

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

Eric Weisstein's World of Mathematics, Positive Definite Matrix

Cf. A114601, A085656.

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

{ 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 */

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

cp's OEIS Frontend

A085657 Number of n X n symmetric positive definite matrices with 2's on the main diagonal and 1's and 0's elsewhere.

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