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

1,3

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.

Links

Table of n, a(n) for n=1..96.

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).

Example

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).

Mathematica

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

Crossrefs

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

Cf. A074930.

Sequence in context: A370221 A098281 A207324 * A103343 A085263 A115092

Adjacent sequences: A352617 A352618 A352619 * A352621 A352622 A352623

Keyword

nonn,tabf

Author

Luca Onnis, Mar 24 2022

Status

approved