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

%I #10 Oct 31 2023 11:14:49

%S 0,4,18,44,100,180,294,432,630,900,1210,1548,2028,2548,3150,3744,4624,

%T 5436,6498,7500,8820,10164,11638,13104,14900,16900,18792,20972,23548,

%U 26100,28830,31360,34848,38148,41650,44676,49284,53428,57798,62000

%N Number of 2 X 2 symmetric matrices over Z(n) having nonzero determinant.

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

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

%t 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 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 *)

%o (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

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

%K nonn,easy

%O 1,2

%A _T. D. Noe_, Jan 12 2006