A115075 Number of 2 X 2 symmetric matrices over Z(n) having determinant 0.

1, 4, 9, 20, 25, 36, 49, 80, 99, 100, 121, 180, 169, 196, 225, 352, 289, 396, 361, 500, 441, 484, 529, 720, 725, 676, 891, 980, 841, 900, 961, 1408, 1089, 1156, 1225, 1980, 1369, 1444, 1521, 2000, 1681, 1764, 1849, 2420, 2475, 2116, 2209, 3168, 2695, 2900

Offset: 1

Author

T. D. Noe, Jan 12 2006

Keywords

• mult
• nonn
• easy

Crossrefs

Cf. A020478 (number of singular 2 X 2 matrices over Z(n)), A115077.

Cf. A002117, A013662.

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] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]  (* Amiram Eldar, Oct 31 2023 *)

PARI

a(n) = {my(f = factor(n), p, e); 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 - A115077(n).

For squarefree n, a(n) = n^2.

Multiplicative with a(p^e) = p^(e)*(p^(e)+p^(e-1)-p^(ceiling(e/2)-1)).

From Amiram Eldar, Oct 31 2023: (Start)

Dirichlet g.f.: zeta(s-2) * zeta(2*s-3) / zeta(2*s-2).

Sum_{k=1..n} a(k) ~ (zeta(3)/(3*zeta(4))) * n^3. (End)