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A264083 Number of orthogonal 3 X 3 matrices over the ring Z/nZ. 7


%S 1,6,48,384,240,288,672,6144,1296,1440,2640,18432,4368,4032,11520,

%T 49152,9792,7776,13680,92160,32256,15840,24288,294912,30000,26208,

%U 34992,258048,48720,69120,59520,393216,126720,58752,161280,497664,101232,82080,209664,1474560

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

%C 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).

%C For n > 1, a(n) is divisible by 6*A060594(n)^3. - _Robert Israel_, Dec 16 2015

%H Robert Israel, <a href="/A264083/b264083.txt">Table of n, a(n) for n = 1..124</a>

%F For p an odd prime, a(p) = 2*p*(p^2-1). - _Tom Edgar_, Nov 04 2015

%F From _Robert Israel_, Dec 16 2015: (Start)

%F Conjectures:

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

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

%p F:= proc(n) local R,V,nR,S,nS,Rp,nRp,i,j,a,b,c,t,r,r1,count;

%p 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)]);

%p nR:= nops(R);

%p S:= select(t -> t^2 mod n = 1, {$2..n-1});

%p nS:= nops(S);

%p for r in R do if not assigned(V[r]) then

%p for c in S do V[c*r mod n] := 0 od

%p fi od;

%p R:= select(r -> not assigned(V[r]), R);

%p nR:= nops(R);

%p count:= 0;

%p for i from 1 to nR do

%p r:= R[i];

%p Rp:= select(j -> R[j][1]*r[1] + R[j][2]*r[2] + R[j][3]*r[3] mod n = 0, [$i+1..nR]);

%p nRp:= nops(Rp);

%p for j from 1 to nRp do

%p r1:= R[Rp[j]];

%p 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]));

%p od

%p od;

%p count;

%p end proc:

%p F(1):= 1:

%p seq(F(n), n=1..40); # _Robert Israel_, Dec 16 2015

%o (MAGMA)

%o Enter R := IntegerRing(n);

%o korthmat := function(R,n,k);

%o O := [];

%o M := MatrixAlgebra(R,n);

%o for x in M do

%o if x*Transpose(x) eq k*M!1 and Transpose(x)*x eq k*M!1 then

%o O := Append(O,x);

%o end if;

%o end for;

%o return O;

%o end function;

%o # korthmat(R,3,1);

%o (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

%Y Cf. A060594, A087784, A208895.

%K nonn,mult

%O 1,2

%A _Charles Repizo_, Nov 03 2015

%E a(11)-a(31) from _Tom Edgar_, Nov 05 2015

%E a(31) corrected by _Robert Israel_, Dec 15 2015

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Last modified August 2 21:30 EDT 2021. Contains 346429 sequences. (Running on oeis4.)