A352620 - OEIS

%I #62 Mar 27 2022 13:43:53

%S 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,

%T 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,

%U 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

%N 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.

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

%C Those matrices are symmetric.

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

%H Onno Cain, <a href="https://arxiv.org/abs/1908.03236">Gaussian Integers, Rings, Finite Fields, and the Magic Square of Squares</a>, arXiv:1908.03236 [math.RA], 2019.

%H Matt Parker and Brady Haran, <a href="https://www.youtube.com/watch?v=FCczHiXPVcA&amp;t=15s">Finite Fields & Return of The Parker Square</a>, Numberphile video (Oct 7, 2021).

%e Matrices begin:

%e   n=1:  1,

%e   n=2:  1, 2,

%e         2, 1,

%e   n=3:  1, 2, 3,

%e         2, 0, 2,

%e         3, 2, 1,

%e   n=4:  1, 2, 3, 4,

%e         2, 4, 1, 3,

%e         3, 1, 4, 2,

%e         4, 3, 2, 1;

%e For example, the 6 X 6 matrix generated by Z_7 is the following:

%e   1 2 3 4 5 6

%e   2 4 6 1 3 5

%e   3 6 2 5 1 4

%e   4 1 5 2 6 3

%e   5 3 1 6 4 2

%e   6 5 4 3 2 1

%e The trace of this matrix is 14 = A048153(7).

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

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

%Y Cf. A074930.

%K nonn,tabf

%O 1,3

%A _Luca Onnis_, Mar 24 2022