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A194894 The number of the ordered triples (A,B,C) satisfying the system of the modular relations {A*B - B*A = C, B*C - C*B = A, C*A - A*C = B}, where A,B,C are distinct 2 X 2 matrices over Z(n).

%I A194894 #27 Jan 12 2017 02:05:20
%S A194894 0,0,24,0,120,24,336,0,648,120,1320,24,2184,336,3024,0,4896,648,6840,
%T A194894 120,8424,1320,12144,24,15000,2184,17496,336,24360,3024,29760,0,33024,
%U A194894 4896,40776,648,50616,6840,54624,120,68880
%N A194894 The number of the ordered triples (A,B,C) satisfying the system of the modular relations {A*B - B*A = C, B*C - C*B = A, C*A - A*C = B}, where A,B,C are distinct 2 X 2 matrices over Z(n).
%C A194894 If (A,B,C) is a triple and X is chosen from among A,B,C, then trace(X)=0 mod n, X*X = -det(X)*IdentityMatrix mod n, A*B + B*A = B*C + C*B = C*A + A*C = 0 mod n, det(A) = det(B) = det(C) mod n, A*A = B*B = C*C mod n, A = 2*B*C, B = 2*C*A, C = 2*A*B mod n.
%C A194894 For a given value of n, consider the family of triples (A,B,C) for which d = det(A) = det(B) = det(C) mod n. Let b(n,d) denote the number of elements of the set {A: (A,B,C) is a triple and det(A) = d}. Let b(n) = Sum{ b(n,d) for all such d }, for example, d(15) = 6 + 30 + 180. Detailed results of searching for trios (N(d) = number of triples in the family):
%C A194894 . .n b(n,d) ...d ......N
%C A194894 . .1 .....0 .... ......0
%C A194894 . .2 .....0 .... ......0
%C A194894 . .3 .....6 ...1 .....24
%C A194894 . .4 .....0 .... ......0
%C A194894 . .5 ....30 ...4 ....120
%C A194894 . .6 .....6 ...4 .....24
%C A194894 . .7 ....42 ...2 ....336
%C A194894 . .8 .....0 .... ......0
%C A194894 . .9 ....54 ...7 ....648
%C A194894 . 10 ....30 ...4 ....120
%C A194894 . 11 ...110 ...3 ...1320
%C A194894 . 12 .....6 ...4 .....24
%C A194894 . 13 ...182 ..10 ...2184
%C A194894 . 14 ....42 ...2 ....336
%C A194894 . 15......6 ..10 .....24
%C A194894 . 15.....30 ...9 ....120
%C A194894 . 15....180 ...4 ...2880
%C A194894 . 16 .....0 .... ......0
%C A194894 . 17 ...306 ..13 ...4896
%C A194894 . 18 ....54 ..16 ....648
%C A194894 . 19 ...342 ...5 ...6840
%C A194894 . 20 ....30 ...4 ....120
%C A194894 . 21......6 ...7 .....24
%C A194894 . 21....252 ..16 ...8064
%C A194894 . 21.....42 ...9 ....336
%C A194894 . 22 ...110 ..14 ...1320
%C A194894 . 23 ...506 ...6 ..12144
%C A194894 . 24 .....6 ..16 .....24
%C A194894 . 25 ...750 ..19 ..15000
%C A194894 . 26 ...182 .... ...2184
%C A194894 . 27 ...486 ...7 ..17496
%C A194894 . 28 ....42 ..16 ....336
%C A194894 . 29 ...870 ..22 ..24360
%C A194894 . 30......6 ..10 .....24
%C A194894 . 30.....30 ..24 ....120
%C A194894 . 30....180 ...4 ...2880
%C A194894 . 31 ...930 ...8 ..29760
%C A194894 . 32 .....0 .... ......0
%C A194894 . 33......6 ..22 .....24
%C A194894 . 33....660 ..25 ..31680
%C A194894 . 33....110 ...3 ...1320
%C A194894 . 34 ...306 ..30 ...4896
%C A194894 . 35...1260 ...9 ..40320
%C A194894 . 35.....42 ..30 ....336
%C A194894 . 35.....30 ..14 ....120
%C A194894 . 36 ....54 ..16 ....648
%C A194894 . 37 ..1406 ..28 ..50616
%C A194894 . 38 ...342 ..24 ...6840
%C A194894 . 39......6 ..13 .....24
%C A194894 . 39....182 ..36 ...2184
%C A194894 . 39...1092 ..10 ..52416
%C A194894 . 40 ....30 ..24 ....120
%C A194894 . 41 ..1722 ..31 ..68880
%C A194894 Remarks for the cases n<=41 (conjectures for n>41):
%C A194894 b(n) is similar to a(n), i.e., b(2^e)=0 for e>=0, b(m*2^e)=b(m) for m>=0 and e>=0, b(m*n) = b(m) + b(n) + b(m)*b(n) for gcd(m,n)=1;
%C A194894 b(p) = (p-1)*p for primes of the form p = 4*k + 1;
%C A194894 b(p) = p*(p+1) for primes of the form p = 4*k - 1;
%C A194894 b(p^e) = b(p)*(p^(2*(e-1))) for odd primes p and e>=1;
%C A194894 if n=p^e (p is odd prime, e>=1) then d is a constant for all trios (there is only one family), moreover 4*d=1 (mod n).
%F A194894 a(2^e) = 0 for e>=0; a( m*(2^e) ) = a(m) for m>=1,e>=0.
%F A194894 a(p^e) = (p^2-1)*p^(3*e-2) for odd prime p,e>=1.
%F A194894 a(m*n) = a(m) + a(n) + a(m)*a(n) for gcd(m,n)=1
%e A194894 The matrices A=[0,1;2,0], B=[1,1;1,2], C=[2,1;1,1] of row order form satisfy the system of the (mod 3)-relations {A*B - B*A = C, A#B, B*C - C*B = A, B#C, C*A - A*C = B, C#A}, so we have a trio (+A,+B,+C). All the solutions of the system can be represented by the trios
%e A194894 (+A,+B,+C), (+B,+C,+A), (+C,+A,+B),
%e A194894 (+A,-C,+B), (-C,+B,+A), (+B,+A,-C),
%e A194894 (+A,+C,-B), (+C,-B,+A), (-B,+A,+C),
%e A194894 (+A,-B,-C), (-B,-C,+A), (-C,+A,-B),
%e A194894 (-A,+B,-C), (+B,-C,-A), (-C,-A,+B),
%e A194894 (-A,-C,-B), (-C,-B,-A), (-B,-A,-C),
%e A194894 (-A,+C,+B), (+C,+B,-A), (+B,-A,+C),
%e A194894 (-A,-B,+C), (-B,+C,-A), (+C,-A,-B), so a(3)=24.
%Y A194894 Cf. A000056, A181107.
%K A194894 easy,nonn
%O A194894 1,3
%A A194894 _Erdos Pal_, Sep 04 2011