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

%I A115075 #10 Oct 31 2023 11:14:53
%S A115075 1,4,9,20,25,36,49,80,99,100,121,180,169,196,225,352,289,396,361,500,
%T A115075 441,484,529,720,725,676,891,980,841,900,961,1408,1089,1156,1225,1980,
%U A115075 1369,1444,1521,2000,1681,1764,1849,2420,2475,2116,2209,3168,2695,2900
%N A115075 Number of 2 X 2 symmetric matrices over Z(n) having determinant 0.
%F A115075 a(n) = n^3 - A115077(n).
%F A115075 For squarefree n, a(n) = n^2.
%F A115075 Multiplicative with a(p^e) = p^(e)*(p^(e)+p^(e-1)-p^(ceiling(e/2)-1)).
%F A115075 From _Amiram Eldar_, Oct 31 2023: (Start)
%F A115075 Dirichlet g.f.: zeta(s-2) * zeta(2*s-3) / zeta(2*s-2).
%F A115075 Sum_{k=1..n} a(k) ~ (zeta(3)/(3*zeta(4))) * n^3. (End)
%t A115075 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}]
%t A115075 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 *)
%o A115075 (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
%Y A115075 Cf. A020478 (number of singular 2 X 2 matrices over Z(n)), A115077.
%Y A115075 Cf. A002117, A013662.
%K A115075 mult,nonn,easy
%O A115075 1,2
%A A115075 _T. D. Noe_, Jan 12 2006