A181107 -id:A181107 - cp's OEIS frontend

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

0, 0, 24, 0, 120, 24, 336, 0, 648, 120, 1320, 24, 2184, 336, 3024, 0, 4896, 648, 6840, 120, 8424, 1320, 12144, 24, 15000, 2184, 17496, 336, 24360, 3024, 29760, 0, 33024, 4896, 40776, 648, 50616, 6840, 54624, 120, 68880

Offset: 1

Erdos Pal, Sep 04 2011

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.
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):
. .n b(n,d) ...d ......N
. .1 .....0 .... ......0
. .2 .....0 .... ......0
. .3 .....6 ...1 .....24
. .4 .....0 .... ......0
. .5 ....30 ...4 ....120
. .6 .....6 ...4 .....24
. .7 ....42 ...2 ....336
. .8 .....0 .... ......0
. .9 ....54 ...7 ....648
. 10 ....30 ...4 ....120
. 11 ...110 ...3 ...1320
. 12 .....6 ...4 .....24
. 13 ...182 ..10 ...2184
. 14 ....42 ...2 ....336
. 15......6 ..10 .....24
. 15.....30 ...9 ....120
. 15....180 ...4 ...2880
. 16 .....0 .... ......0
. 17 ...306 ..13 ...4896
. 18 ....54 ..16 ....648
. 19 ...342 ...5 ...6840
. 20 ....30 ...4 ....120
. 21......6 ...7 .....24
. 21....252 ..16 ...8064
. 21.....42 ...9 ....336
. 22 ...110 ..14 ...1320
. 23 ...506 ...6 ..12144
. 24 .....6 ..16 .....24
. 25 ...750 ..19 ..15000
. 26 ...182 .... ...2184
. 27 ...486 ...7 ..17496
. 28 ....42 ..16 ....336
. 29 ...870 ..22 ..24360
. 30......6 ..10 .....24
. 30.....30 ..24 ....120
. 30....180 ...4 ...2880
. 31 ...930 ...8 ..29760
. 32 .....0 .... ......0
. 33......6 ..22 .....24
. 33....660 ..25 ..31680
. 33....110 ...3 ...1320
. 34 ...306 ..30 ...4896
. 35...1260 ...9 ..40320
. 35.....42 ..30 ....336
. 35.....30 ..14 ....120
. 36 ....54 ..16 ....648
. 37 ..1406 ..28 ..50616
. 38 ...342 ..24 ...6840
. 39......6 ..13 .....24
. 39....182 ..36 ...2184
. 39...1092 ..10 ..52416
. 40 ....30 ..24 ....120
. 41 ..1722 ..31 ..68880
Remarks for the cases n<=41 (conjectures for n>41):
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;
b(p) = (p-1)*p for primes of the form p = 4*k + 1;
b(p) = p*(p+1) for primes of the form p = 4*k - 1;
b(p^e) = b(p)*(p^(2*(e-1))) for odd primes p and e>=1;
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).

			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
(+A,+B,+C), (+B,+C,+A), (+C,+A,+B),
(+A,-C,+B), (-C,+B,+A), (+B,+A,-C),
(+A,+C,-B), (+C,-B,+A), (-B,+A,+C),
(+A,-B,-C), (-B,-C,+A), (-C,+A,-B),
(-A,+B,-C), (+B,-C,-A), (-C,-A,+B),
(-A,-C,-B), (-C,-B,-A), (-B,-A,-C),
(-A,+C,+B), (+C,+B,-A), (+B,-A,+C),
(-A,-B,+C), (-B,+C,-A), (+C,-A,-B), so a(3)=24.

Cf. A000056, A181107.

a(2^e) = 0 for e>=0; a( m*(2^e) ) = a(m) for m>=1,e>=0.
a(p^e) = (p^2-1)*p^(3*e-2) for odd prime p,e>=1.
a(m*n) = a(m) + a(n) + a(m)*a(n) for gcd(m,n)=1

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

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