A352620 Irregular triangle read by rows which are rows of successive n X n matrices M(n) with entries M(n)[i,j] = i*j mod n+1.

1, 1, 2, 2, 1, 1, 2, 3, 2, 0, 2, 3, 2, 1, 1, 2, 3, 4, 2, 4, 1, 3, 3, 1, 4, 2, 4, 3, 2, 1, 1, 2, 3, 4, 5, 2, 4, 0, 2, 4, 3, 0, 3, 0, 3, 4, 2, 0, 4, 2, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 2, 4, 6, 1, 3, 5, 3, 6, 2, 5, 1, 4, 4, 1, 5, 2, 6, 3, 5, 3, 1, 6, 4, 2, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5

Offset: 1

Author

Luca Onnis, Mar 24 2022

Keywords

• nonn
• tabf

Comments

Each matrix represents all possible products between the elements of Z_(n+1), where Z_k is the ring of integers mod k.

Those matrices are symmetric.

The first row is equal to the first column which is equal to 1,2,...,n.

Examples

			Matrices begin:
  n=1:  1,
  n=2:  1, 2,
        2, 1,
  n=3:  1, 2, 3,
        2, 0, 2,
        3, 2, 1,
  n=4:  1, 2, 3, 4,
        2, 4, 1, 3,
        3, 1, 4, 2,
        4, 3, 2, 1;
For example, the 6 X 6 matrix generated by Z_7 is the following:
  1 2 3 4 5 6
  2 4 6 1 3 5
  3 6 2 5 1 4
  4 1 5 2 6 3
  5 3 1 6 4 2
  6 5 4 3 2 1
The trace of this matrix is 14 = A048153(7).
		

Links

• Onno Cain, Gaussian Integers, Rings, Finite Fields, and the Magic Square of Squares, arXiv:1908.03236 [math.RA], 2019.
• Matt Parker and Brady Haran, Finite Fields & Return of The Parker Square, Numberphile video (Oct 7, 2021).

Crossrefs

Cf. A048153 (traces), A349099 (permanents), A160255 (sum entries), A088922 (ranks).

Cf. A074930.

Programs

Mathematica

Flatten[Table[Table[Mod[k*Table[i, {i, 1, p - 1}], p], {k, 1, p - 1}], {p, 1, 10}]]