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A264083 Number of orthogonal 3 X 3 matrices over the ring Z/nZ. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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)

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

PROG

(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);

(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

CROSSREFS

Cf. A060594, A087784, A208895.

Sequence in context: A291033 A155130 A250164 * A083233 A002918 A005399

Adjacent sequences:  A264080 A264081 A264082 * A264084 A264085 A264086

KEYWORD

nonn,mult

AUTHOR

Charles Repizo, Nov 03 2015

EXTENSIONS

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

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

STATUS

approved

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Last modified June 14 15:44 EDT 2021. Contains 345025 sequences. (Running on oeis4.)