A115225 Number of 3 X 3 symmetric matrices over Z(n) having nonzero determinant.

0, 28, 468, 2688, 12400, 37260, 100548, 215040, 454896, 883900, 1609300, 2618496, 4453488, 6913900, 10548900, 15138816, 22713088, 31256604, 44563284, 59459200, 81302760, 107538508, 141587908, 178808832, 232500000, 295476220, 368465760

Offset: 1

Author

T. D. Noe, Jan 16 2006

Keywords

• nonn

Crossrefs

Cf. A115222 (number of 3 X 3 matrices over Z(n) having nonzero determinant).

Programs

Mathematica

Table[cnt=0; Do[m={{a, b, c}, {b, d, e}, {c, e, f}}; If[Det[m, Modulus->n]>0, cnt++ ], {a, 0, n-1}, {b, 0, n-1}, {c, 0, n-1}, {d, 0, n-1}, {e, 0, n-1}, {f, 0, n-1}]; cnt, {n, 2, 20}]

Formula

a(n)=n^6-A115223(n). For prime n, a(n)=(n^3-1)(n-1)n^2.