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

1,2

Links

Table of n, a(n) for n=1..40.

Formula

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

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A126283 A023618 A366985 * A278046 A258634 A280599

Adjacent sequences: A115074 A115075 A115076 * A115078 A115079 A115080

Keyword

nonn,easy

Author

T. D. Noe, Jan 12 2006

Status

approved