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
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