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A204016 Symmetric matrix based on f(i,j) = max(j mod i, i mod j), by antidiagonals.

%I A204016 #20 Oct 24 2024 05:40:47
%S A204016 0,1,1,1,0,1,1,2,2,1,1,2,0,2,1,1,2,3,3,2,1,1,2,3,0,3,2,1,1,2,3,4,4,3,
%T A204016 2,1,1,2,3,4,0,4,3,2,1,1,2,3,4,5,5,4,3,2,1,1,2,3,4,5,0,5,4,3,2,1,1,2,
%U A204016 3,4,5,6,6,5,4,3,2,1,1,2,3,4,5,6,0,6,5,4,3,2,1,1,2,3,4,5,6,7,7
%N A204016 Symmetric matrix based on f(i,j) = max(j mod i, i mod j), by antidiagonals.
%C A204016 A204016 represents the matrix M given by f(i,j) = max{(j mod i), (i mod j)} for i >= 1 and j >= 1.  See A204017 for characteristic polynomials of principal submatrices of M, with interlacing zeros.
%C A204016 Guide to symmetric matrices M based on functions f(i,j) and characteristic polynomial sequences (c.p.s.) with interlaced zeros:
%C A204016 f(i,j)..........................M.........c.p.s.
%C A204016 C(i+j,j)........................A007318...A045912
%C A204016 min(i,j)........................A003983...A202672
%C A204016 max(i,j)........................A051125...A203989
%C A204016 (i+j)*min(i,j)..................A203990...A203991
%C A204016 |i-j|...........................A049581...A203993
%C A204016 max(i-j+1,j-i+1)................A143182...A203992
%C A204016 min(i-j+1,j-i+1)................A203994...A203995
%C A204016 min(i(j+1),j(i+1))..............A203996...A203997
%C A204016 max(i(j+1)-1,j(i+1)-1)..........A203998...A203999
%C A204016 min(i(j+1)-1,j(i+1)-1)..........A204000...A204001
%C A204016 min(2i+j,i+2j)..................A204002...A204003
%C A204016 max(2i+j-2,i+2j-2)..............A204004...A204005
%C A204016 min(2i+j-2,i+2j-2)..............A204006...A204007
%C A204016 max(3i+j-3,i+3j-3)..............A204008...A204011
%C A204016 min(3i+j-3,i+3j-3)..............A204012...A204013
%C A204016 min(3i-2,3j-2)..................A204028...A204029
%C A204016 1+min(j mod i, i mod j).........A204014...A204015
%C A204016 max(j mod i, i mod j)...........A204016...A204017
%C A204016 1+max(j mod i, i mod j).........A204018...A204019
%C A204016 min(i^2,j^2)....................A106314...A204020
%C A204016 min(2i-1, 2j-1).................A157454...A204021
%C A204016 max(2i-1, 2j-1).................A204022...A204023
%C A204016 min(i(i+1)/2,j(j+1)/2)..........A106255...A204024
%C A204016 gcd(i,j)........................A003989...A204025
%C A204016 gcd(i+1,j+1)....................A204030...A204111
%C A204016 min(F(i+1),F(j+1)),F=A000045....A204026...A204027
%C A204016 gcd(F(i+1),F(j+1)),F=A000045....A204112...A204113
%C A204016 gcd(L(i),L(j)),L=A000032........A204114...A204115
%C A204016 gcd(2^i-1,2^j-2)................A204116...A204117
%C A204016 gcd(prime(i),prime(j))..........A204118...A204119
%C A204016 gcd(prime(i+1),prime(j+1))......A204120...A204121
%C A204016 gcd(2^(i-1),2^(j-1))............A144464...A204122
%C A204016 max(floor(i/j),floor(j/i))......A204123...A204124
%C A204016 min(ceiling(i/j),ceiling(j/i))..A204143...A204144
%C A204016 Delannoy matrix.................A008288...A204135
%C A204016 max(2i-j,2j-i)..................A204154...A204155
%C A204016 -1+max(3i-j,3j-i)...............A204156...A204157
%C A204016 max(3i-2j,3j-2i)................A204158...A204159
%C A204016 floor((i+1)/2)..................A204164...A204165
%C A204016 ceiling((i+1)/2)................A204166...A204167
%C A204016 i+j.............................A003057...A204168
%C A204016 i+j-1...........................A002024...A204169
%C A204016 i*j.............................A003991...A204170
%C A204016 ..abbreviation below:  AOE means "all 1's except"
%C A204016 AOE f(i,i)=i....................A204125...A204126
%C A204016 AOE f(i,i)=A000045(i+1).........A204127...A204128
%C A204016 AOE f(i,i)=A000032(i)...........A204129...A204130
%C A204016 AOE f(i,i)=2i-1.................A204131...A204132
%C A204016 AOE f(i,i)=2^(i-1)..............A204133...A204134
%C A204016 AOE f(i,i)=3i-2.................A204160...A204161
%C A204016 AOE f(i,i)=floor((i+1)/2).......A204162...A204163
%C A204016 ...
%C A204016 Other pairs (M, c.p.s.): (A204171, A204172) to (A204183, A204184)
%C A204016 See A202695 for a guide to choices of symmetric matrix M for which the zeros of the characteristic polynomials are all positive.
%e A204016 Northwest corner:
%e A204016   0 1 1 1 1 1 1 1
%e A204016   0 1 2 2 2 2 2 2
%e A204016   1 2 0 3 3 3 3 3
%e A204016   1 2 3 0 4 4 4 4
%e A204016   1 2 3 4 0 5 5 5
%e A204016   1 2 3 4 5 0 6 6
%e A204016   1 2 3 4 5 6 0 7
%t A204016 f[i_, j_] := Max[Mod[i, j], Mod[j, i]];
%t A204016 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204016 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204016 Flatten[Table[f[i, n + 1 - i],
%t A204016 {n, 1, 12}, {i, 1, n}]]  (* A204016 *)
%t A204016 p[n_] := CharacteristicPolynomial[m[n], x];
%t A204016 c[n_] := CoefficientList[p[n], x]
%t A204016 TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
%t A204016 Table[c[n], {n, 1, 12}]
%t A204016 Flatten[%]               (* A204017 *)
%t A204016 TableForm[Table[c[n], {n, 1, 10}]]
%Y A204016 Cf. A204017, A202453.
%K A204016 nonn,tabl
%O A204016 1,8
%A A204016 _Clark Kimberling_, Jan 10 2012