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).

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

1,3

Comments

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).

Links

Table of n, a(n) for n=1..41.

Formula

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

Example

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.

Crossrefs

Cf. A000056, A181107.

Sequence in context: A362795 A242837 A353324 * A364226 A128379 A111983

Adjacent sequences: A194891 A194892 A194893 * A194895 A194896 A194897

Keyword

easy,nonn

Author

Erdos Pal, Sep 04 2011

Status

approved