A160255 -id:A160255 - cp's OEIS frontend

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

Luca Onnis, Mar 24 2022

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.

			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).

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).

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

Cf. A074930.

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

A349099 a(n) is the permanent of the n X n matrix M(n) defined as M(n)[i,j] = i*j (mod n + 1).

Original entry on oeis.org

1, 1, 5, 32, 1074, 12600, 1525292, 34078720, 4072850100, 263459065600, 106809546673488, 2254519427530752, 3172225081523720416, 210351382651302645760, 45654014718074873700000, 11122845097194072534155264, 18156837198112938091803999360, 795289872611524024920215715840

Offset: 0

Stefano Spezia, Mar 25 2022

nonn

Det(M(n)) = 0 iff n = 4 or n > 5.
Rank(M(n)) = A088922(n+1).
Tr(M(n)) = A048153(n+1).

			See A352620 for the examples of matrix M(n).

Cf. A352620.

Cf. A048153, A074930, A088922, A160255.

a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](
         Matrix(n, (i, j)-> (i*j) mod (n+1)))):
seq(a(n), n=0..16);  # Alois P. Heinz, Mar 25 2022

Join[{1},Table[Permanent[Table[Mod[j*Table[i, {i, n}], n+1], {j, n}]], {n, 17}]]

a(n) = matpermanent(matrix(n,n,i,j,(i*j)%(n+1))); \\ Michel Marcus, Mar 26 2022

cp's OEIS Frontend

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.

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