A338099 Number of pairs of 2 X 2 matrices (X,Y) over Z/nZ such that X*Y = 0 and Y*X <> 0.

0, 18, 192, 1296, 2880, 15186, 16128, 62208, 88128, 199890, 158400, 764688, 366912, 1063314, 1551360, 2506752, 1410048, 5742738, 2462400, 9461520, 8089536, 9973458, 6412032, 31593216, 14040000, 22817106, 27713664, 48947472, 20462400, 97370130, 28569600, 92012544

Offset

1,2

Links

Table of n, a(n) for n=1..32.

Mathematica

WW = Array[W, {2, 2}];
Ma[n_] :=  Ma[n] = Mod[Flatten[Table[ WW, {W[1, 1], n}, {W[1, 2], n}, {W[2, 1], n}, {W[2, 2], n}], 3], n]
S[n_] := S[n] =   Sum[If[Mod[Ma[n][[i]].Ma[n][[j]], n] == 0  WW && !Mod[ Ma[n][[j]].Ma[n][[i]], n] == 0  WW , 1, 0], {i, n^4}, {j, n ^4}]
Array[S, 9]

Prog

(Python)
from numba import jit
@jit(nopython=True)
def a(n):
  c = 0
  for ax in range(n):
    for bx in range(n):
      for cx in range(bx, n):
        card = 1 + (cx > bx)
        for dx in range(n):
          for ay in range(n):
            for by in range(n):
              for cy in range(n):
                if (ax*ay + bx*cy)%n == 0:
                  if (cx*ay + dx*cy)%n == 0:
                    for dy in range(n):
                      if ax==ay and bx==by and cx==cy and dx==dy: continue
                      if (ax*by + bx*dy)%n == 0:
                        if (cx*by + dx*dy)%n == 0:
                          if (ay*ax + by*cx)%n != 0: c += card; continue
                          if (ay*bx + by*dx)%n != 0: c += card; continue
                          if (cy*ax + dy*cx)%n != 0: c += card; continue
                          if (cy*bx + dy*dx)%n != 0: c += card; continue
  return c
print([a(n) for n in range(1, 12)]) # Michael S. Branicky, Dec 27 2020

Crossrefs

Cf. A227433 (Number of pairs of 2 X 2 matrices over Z/nZ that do not commute).

Sequence in context: A268447 A259163 A004314 * A125406 A318161 A182311

Adjacent sequences: A338096 A338097 A338098 * A338100 A338101 A338102

Keyword

nonn

Author

José María Grau Ribas, Oct 10 2020

Extensions

Three terms corrected by José María Grau Ribas, Dec 19 2020

a(12)-a(32) from Michael S. Branicky, Dec 27 2020

Status

approved