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.

%I A352620 #62 Mar 27 2022 13:43:53
%S A352620 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 A352620 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 A352620 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 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.
%C A352620 Each matrix represents all possible products between the elements of Z_(n+1), where Z_k is the ring of integers mod k.
%C A352620 Those matrices are symmetric.
%C A352620 The first row is equal to the first column which is equal to 1,2,...,n.
%H A352620 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 A352620 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 A352620 Matrices begin:
%e A352620   n=1:  1,
%e A352620   n=2:  1, 2,
%e A352620         2, 1,
%e A352620   n=3:  1, 2, 3,
%e A352620         2, 0, 2,
%e A352620         3, 2, 1,
%e A352620   n=4:  1, 2, 3, 4,
%e A352620         2, 4, 1, 3,
%e A352620         3, 1, 4, 2,
%e A352620         4, 3, 2, 1;
%e A352620 For example, the 6 X 6 matrix generated by Z_7 is the following:
%e A352620   1 2 3 4 5 6
%e A352620   2 4 6 1 3 5
%e A352620   3 6 2 5 1 4
%e A352620   4 1 5 2 6 3
%e A352620   5 3 1 6 4 2
%e A352620   6 5 4 3 2 1
%e A352620 The trace of this matrix is 14 = A048153(7).
%t A352620 Flatten[Table[Table[Mod[k*Table[i, {i, 1, p - 1}], p], {k, 1, p - 1}], {p, 1, 10}]]
%Y A352620 Cf. A048153 (traces), A349099 (permanents), A160255 (sum entries), A088922 (ranks).
%Y A352620 Cf. A074930.
%K A352620 nonn,tabf
%O A352620 1,3
%A A352620 _Luca Onnis_, Mar 24 2022