A115077 Number of 2 X 2 symmetric matrices over Z(n) having nonzero determinant.

0, 4, 18, 44, 100, 180, 294, 432, 630, 900, 1210, 1548, 2028, 2548, 3150, 3744, 4624, 5436, 6498, 7500, 8820, 10164, 11638, 13104, 14900, 16900, 18792, 20972, 23548, 26100, 28830, 31360, 34848, 38148, 41650, 44676, 49284, 53428, 57798, 62000

Offset: 1

Author

T. D. Noe, Jan 12 2006

Keywords

• nonn
• easy

Crossrefs

Cf. A005353 (number of 2 X 2 matrices over Z(n) having nonzero determinant), A115075.

Programs

Mathematica

Table[cnt=0; Do[m={{a, b}, {b, c}}; If[Det[m, Modulus->n]>0, cnt++ ], {a, 0, n-1}, {b, 0, n-1}, {c, 0, n-1}]; cnt, {n, 50}]
f[p_, e_] := p^e*(p^e + p^(e-1) - p^(Ceiling[e/2] - 1)); a[1] = 0; a[n_] := n^3 - Times @@ f @@@ FactorInteger[n]; Array[a, 100]  (* Amiram Eldar, Oct 31 2023 *)

PARI

a(n) = {my(f = factor(n), p, e); n^3 - prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; p^e*(p^e + p^(e-1) - p^((e+1)\2 - 1)));} \\ Amiram Eldar, Oct 31 2023

Formula

a(n) = n^3 - A115075(n).

For squarefree n, a(n) = (n-1)*n^2.