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User: Erdos Pal

Erdos Pal has authored 2 sequences.

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

A181107 Triangle read by rows: T(n,k) is the number of 2 X 2 matrices over Z(n) having determinant congruent to k mod n, 1 <= n, 0 <= k <= n-1.

Original entry on oeis.org

1, 10, 6, 33, 24, 24, 88, 48, 72, 48, 145, 120, 120, 120, 120, 330, 144, 240, 198, 240, 144, 385, 336, 336, 336, 336, 336, 336, 736, 384, 576, 384, 672, 384, 576, 384, 945, 648, 648, 864, 648, 648, 864, 648, 648, 1450, 720, 1200, 720, 1200, 870, 1200, 720, 1200, 720

Offset: 1

Erdos Pal, Oct 03 2010

The n-th row is {T(n,0),T(n,1),...,T(n,n-1)}.
Let m denote the prime power p^e.
T(m,0) = A020478(m) = (p^(e+1) + p^e-1)*p^(2*e-1).
T(m,1) = A000056(m) = (p^2-1)*p^(3*e-2).
T(prime(n),1) = A127917(n).
Sum_{k=1..n-1} T(n,k) = A005353(n).
T(n,1) = n*A007434(n) for n>=1 because A000056(n) = n*Jordan_Function_J_2(n).
T(2^n,1) = A083233(n) = A164640(2n) for n>=1. Proof: a(n):=T(2^n,1); a(1)=6, a(n)=8*a(n-1); A083233(1)=6 and A083233(n) is a geometric series with ratio 8 (because of its g.f.), too; A164640 = {b(1)=1, b(2)=6, b(n)=8*b(n-2)}.
T(2^n,0) = A165148(n) for n>=0, because  2*T(2^n,0) = (3*2^n-1)*4^n.
T(2^e,2) = A003951(e) for 2 <= e. Proof: T(2^e,2) = 9*8^(e-1) is a series with ratio 8 and initial term 72, as A003951(2...inf) is.
Working with consecutive powers of a prime p, we need a definition (0 <= i < e):
N(p^e,i):=#{k: 0 < k < p^e, gcd(k,p^e) = p^i} = (p-1)*p^(e-1-i). We say that these k's belong to i (respect to p^e). Note that N(p^e,0) = EulerPhi(p^e), and if 0 < k < p^e then gcd(k,p^e) = gcd(k,p^(e+1)). Let T(p^e,[i]) denote the common value of T(p^e,k)'s, where k's belong to i (q.v.PROGRAM); for example, T(p^e,[0]) = T(p^e,1). The number of the 2 X 2 matrices over Z(p^e), T(p^e,0) + Sum_{i=0..e-1} T(p^e,[i])*N(p^e,i) = p^(4e) will be useful.
On the hexagon property: Let prime p be given and let T(p^e,[0]), T(p^e,[1]), T(p^e,[2]), ..., T(p^e,[e-2]), T(p^e,[e-1]) form the e-th row of a Pascal-like triangle, e>=1. Let denote X(r,s) an element of the triangle and its value T(p^r,[s]). Let positive integers a and b given, so that the entries A(m-a,n-b), B(m-a,n), C(m,n+a), D(m+b,n+a), E(m+b,n), F(m,n-b) of the triangle form a hexagon spaced around T(p^m,[n]); if a=b=1 then they surround it. If A*C*E = B*D*F, then we say that the triangle T(.,.) has the "hexagon property". (In the case of binomial coefficients X(r,s) = COMB(r,s), the "hexagon property" holds (see [Gupta]) and moreover gcd(A,C,E) = gcd(B,D,F) (see [Hitotumatu & Sato]).)
Corollary 2.2 in [Brent & McKay] says that, for the d X d matrices over Z(p^e), (mutatis mutandis) T_d(p^e,0) = K*(1-P(d+e-1)/P(e-1)) and T_d(p^e,[i]) = K*(q^e)*((1-q^d)/(1-q))*P(d+i-1)/P(i), where q=1/p, K=(p^e)^(d^2), P(t) = Product_{j=1..t} (1-q^j), P(0):=1. (For the case d=2, we have T(p^e,[i]) = (p+1)*(p^(i+1)-1)*p^(3*e-i-2).) Due to [Brent & McKay], it can be simply proved that for d X d matrices the "hexagon property" is true. The formulation implies an obvious generalization: For the entries A(r,u), B(r,v), C(s,w), D(t,w), E(t,v), F(s,u) of the T_d(.,.)-triangle, a hexagon-like property A*C*E = B*D*F holds. This is false in general for the COMB(.,.)-triangle.
Another (rotated-hexagon-like) property: for the entries A(m-b1,n), B(m-a1,n+c2), C(m+a2,n+c2), D(m+b2,n), E(m+a2,n-c1), F(m-a1,n-c1) of the T_d(.,.)-triangle, the property A*C*E = B*D*F holds, if and only if 2*(a1 + a2) = b1 + b2. This is also in general false for COMB(.,.)-triangle.

			From _Andrew Howroyd_, Jul 16 2018: (Start)
Triangle begins:
    1;
   10,   6;
   33,  24,  24;
   88,  48,  72,  48;
  145, 120, 120, 120, 120;
  330, 144, 240, 198, 240, 144;
  385, 336, 336, 336, 336, 336, 336;
  736, 384, 576, 384, 672, 384, 576, 384;
  945, 648, 648, 864, 648, 648, 864, 648, 648;
  ... (End)

Erdos Pal, Rows n=1..100 of triangle, flattened
Richard P. Brent and Brendan D. McKay, Determinants and ranks of random matrices over Z_m, Discrete Mathematics 66 (1987) pp. 35-49.
A. K. Gupta, Generalized hidden hexagon squares, The Fibonacci Quarterly, Vol 12, Number 1, Feb.1974, pp. 45-46.
S. Hitotumatu, D. Sato, Star of David theorem (I), The Fibonacci Quarterly, Vol 13, Number 1, Feb.1975, p. 70.

Column k=0 is A020478.
Column k=1 is A000056.
Row sums are A005353.
Cf. A000252, A127917.

  (* computing T(p^e,k) ; p=prime, 1<=e, 0<=k

S(p,e)={my(u=vector(p^e)); my(t=(p-1)*p^(e-1)); u[1] = p^e + e*t; for(j=1, p^e-1, u[j+1] = t*(1+valuation(j, p))); vector(#u, k, sum(j=0, #u-1, u[j + 1]*u[(j+k-1) % #u + 1]))}
T(n)={my(f=factor(n), v=vector(n,i,1)); for(i=1, #f~, my(r=S(f[i,1], f[i,2])); for(j=0, #v-1, v[j + 1] *= r[j % #r + 1])); v}
for(n=1, 10, print(T(n))); \\ Andrew Howroyd, Jul 16 2018

T(a*b,k) = T(a,(k mod a))*T(b,(k mod b)) if gcd(a,b) = 1.

Sum_{k=1..n-1, gcd(k,n)=1} T(n,k) = A000252(n). - Andrew Howroyd, Jul 16 2018

Terms a(24)-a(55) from b-file by Andrew Howroyd, Jul 16 2018

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User: Erdos Pal

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).

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A181107 Triangle read by rows: T(n,k) is the number of 2 X 2 matrices over Z(n) having determinant congruent to k mod n, 1 <= n, 0 <= k <= n-1.

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cp's OEIS Frontend

User: Erdos Pal

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).

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A181107 Triangle read by rows: T(n,k) is the number of 2 X 2 matrices over Z(n) having determinant congruent to k mod n, 1 <= n, 0 <= k <= n-1.

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Extensions

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).