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A115226
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Order of the group of invertible 3 X 3 symmetric matrices over Z(n).
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0
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1, 28, 468, 1792, 12400, 13104, 100548, 114688, 341172, 347200, 1609300, 838656, 4453488, 2815344, 5803200, 7340032, 22713088, 9552816, 44563284, 22220800, 47056464, 45060400, 141587908, 53673984, 193750000, 124697664, 248714388
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Note that A115225 gives the number of 3 x 3 symmetric matrices having nonzero determinant. However, for composite n, a nonzero determinant is not sufficient for the matrix to be invertible; the determinant must also be relatively prime to n.
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FORMULA
| For prime n, a(n)=(n^3-1)(n-1)n^2. In general, a(n)= A115224(n) phi(n)=A064767(n)/A000056(n).
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MATHEMATICA
| Table[cnt=0; Do[m={{a, b, c}, {b, d, e}, {c, e, f}}; If[Det[m, Modulus->n]>0 && MatrixQ[Inverse[m, Modulus->n]], 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}]
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CROSSREFS
| Cf. A000056 (order of the group SL(2, Z_n)), A064767 (order of the group GL(3, Z_n)).
Sequence in context: A000771 A079518 A160060 * A086782 A115225 A140107
Adjacent sequences: A115223 A115224 A115225 * A115227 A115228 A115229
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KEYWORD
| mult,nonn
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AUTHOR
| T. D. Noe (noe(AT)sspectra.com), Jan 16 2006
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