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

1,2

Links

Table of n, a(n) for n=1..50.

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)

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 *)

Prog

(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

Crossrefs

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

Cf. A002117, A013662.

Sequence in context: A308482 A136769 A351600 * A288102 A288100 A063454

Adjacent sequences: A115072 A115073 A115074 * A115076 A115077 A115078

Keyword

mult,nonn,easy

Author

T. D. Noe, Jan 12 2006

Status

approved