A115221 -id:A115221 - cp's OEIS frontend

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

0, 168, 11232, 161280, 1488000, 7170552, 33784128, 106659840, 327525120, 839997000, 2124276000, 4307378688, 9726417792, 18401146008, 34512588000

Offset: 1

T. D. Noe, Jan 16 2006

nonn

See A064767 for the number of invertible matrices.

Table[cnt=0; Do[m={{a, b, c}, {d, e, f}, {g, h, i}}; 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}, {g, 0, n-1}, {h, 0, n-1}, {i, 0, n-1}]; cnt, {n, 2, 7}]

a(n)=n^9-A115221(n). For prime n, a(n)=A064767(n)=n^3 (n-1)^3 (n+1) (n^2+n+1)

A115223 Number of 3 X 3 symmetric matrices over Z(n) having determinant 0.

Original entry on oeis.org

1, 36, 261, 1408, 3225, 9396, 17101, 47104, 76545, 116100, 162261, 367488, 373321, 615636, 841725, 1638400, 1424481, 2755620, 2482597, 4540800, 4463361, 5841396, 6447981, 12294144, 11640625, 13439556, 18954729, 24078208, 20534697, 30302100

Offset: 1

T. D. Noe, Jan 16 2006

Cf. A115221 (number of 3 X 3 matrices over Z(n) having determinant 0).

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}]

a(n)=n^6-A115225(n). For prime n, a(n)=(n^3+n-1)n^2.

cp's OEIS Frontend

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

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