A264083 Number of orthogonal 3 X 3 matrices over the ring Z/nZ.

1, 6, 48, 384, 240, 288, 672, 6144, 1296, 1440, 2640, 18432, 4368, 4032, 11520, 49152, 9792, 7776, 13680, 92160, 32256, 15840, 24288, 294912, 30000, 26208, 34992, 258048, 48720, 69120, 59520, 393216, 126720, 58752, 161280, 497664, 101232, 82080, 209664, 1474560

Offset: 1

Author

Charles Repizo, Nov 03 2015

Keywords

• nonn
• mult

Comments

Number of matrices M = [a,b,c; d,e,f; g,h,i] with 0 <= a, b, c, d, e, f, g, h, i < n such that M*transpose(M) == [1,0,0; 0,1,0; 0,0,1] (mod n).

For n > 1, a(n) is divisible by 6*A060594(n)^3. - Robert Israel, Dec 16 2015

Links

• Robert Israel, Table of n, a(n) for n = 1..124

Crossrefs

Cf. A060594, A087784, A208895.

Programs

Magma

Enter R := IntegerRing(n);
korthmat := function(R,n,k);
O := [];
M := MatrixAlgebra(R,n);
for x in M do
if x*Transpose(x) eq k*M!1 and Transpose(x)*x eq k*M!1 then
O := Append(O,x);
end if;
end for;
return O;
end function;
# korthmat(R,3,1);

Maple

F:= proc(n) local R,V,nR,S,nS,Rp,nRp,i,j,a,b,c,t,r,r1,count;
      R:= select(t -> t[1]^2 + t[2]^2 + t[3]^2 mod n = 1, [seq(seq(seq([a,b,c],a=0..n-1),b=0..n-1),c=0..n-1)]);
      nR:= nops(R);
      S:= select(t -> t^2 mod n = 1, {$2..n-1});
      nS:= nops(S);
      for r in R do if not assigned(V[r]) then
         for c in S do V[c*r mod n] := 0 od
      fi od;
      R:= select(r -> not assigned(V[r]), R);
      nR:= nops(R);
      count:= 0;
      for i from 1 to nR do
        r:= R[i];
        Rp:= select(j -> R[j][1]*r[1] + R[j][2]*r[2] + R[j][3]*r[3] mod n = 0, [$i+1..nR]);
        nRp:= nops(Rp);
        for j from 1 to nRp do
            r1:= R[Rp[j]];
            count:= count + 6*(1+nS)^3*nops(select(k -> R[Rp[k]][1]*r1[1] + R[Rp[k]][2]*r1[2]+R[Rp[k]][3]*r1[3] mod n = 0, [$j+1..nRp]));
        od
      od;
      count;
end proc:
F(1):= 1:
seq(F(n), n=1..40); # Robert Israel, Dec 16 2015

PARI

my(t=Mod(matid(3), n)); sum(a=1, n, sum(b=1, n, sum(c=1, n, sum(d=1, n, sum(e=1, n, sum(f=1, n, sum(g=1, n, sum(h=1, n, sum(i=1, n, my(M=[a, b, c; d, e, f; g, h, i]); M*M~==t))))))))) \\ Charles R Greathouse IV, Nov 10 2015

Formula

For p an odd prime, a(p) = 2*p*(p^2-1). - Tom Edgar, Nov 04 2015

From Robert Israel, Dec 16 2015: (Start)

Conjectures:

a(2^k) = 12*8^k for k >= 3.

For odd primes p, a(p^k) = a(p)*p^(3k-3) for k>=1. (End)

Extensions

a(11)-a(31) from Tom Edgar, Nov 05 2015

a(31) corrected by Robert Israel, Dec 15 2015