|
|
A115075
|
|
Number of 2 X 2 symmetric matrices over Z(n) having determinant 0.
|
|
2
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
LINKS
|
|
|
FORMULA
|
For squarefree n, a(n) = n^2.
Multiplicative with a(p^e) = p^(e)*(p^(e)+p^(e-1)-p^(ceiling(e/2)-1)).
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
|
|
|
KEYWORD
|
mult,nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|