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A204016
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Symmetric matrix based on f(i,j)=max{j mod i, i mod j), by antidiagonals.
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74
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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, 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, 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
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OFFSET
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1,8
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COMMENTS
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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.
Guide to symmetric matrices M based on functions f(i,j) and characteristic polynomial sequences (c.p.s.) with interlaced zeros:
f(i,j).......................M.........c.p.s.
C(i+j,j).....................A007318...A045912
min(i,j).....................A003983...A202672
max(i,j).....................A051125...A203989
(i+j)*min(i,j)...............A203990...A203991
|i-j|........................A049581...A203993
max(i-j+1,j-i+1).............A143182...A203992
min(i-j+1,j-i+1).............A203994...A203995
min(i(j+1),j(i+1))...........A203996...A203997
max(i(j+1)-1,j(i+1)-1).......A203998...A203999
min(i(j+1)-1,j(i+1)-1).......A204000...A204001
min(2i+j,i+2j)...............A204002...A204003
max(2i+j-2,i+2j-2)...........A204004...A204005
min(2i+j-2,i+2j-2)...........A204006...A204007
max(3i+j-3,i+3j-3)...........A204008...A204011
min(3i+j-3,i+3j-3)...........A204012...A204013
min(3i-2,3j-2)...............A204028...A204029
1+min(j mod i, i mod j)......A204014...A204015
max(j mod i, i mod j)........A204016...A204017
1+max(j mod i, i mod j)......A204018...A204019
min(i^2,j^2).................A106314...A204020
min(2i-1, 2j-1)..............A157454...A204021
max(2i-1, 2j-1)..............A204022...A204023
min(i(i+1)/2,j(j+1)/2).......A106255...A204024
GCD(i,j).....................A003989...A204025
GCD(i+1,j+1).................A204030...A204111
min(F(i+1),F(j+1),F=A000045..A204026...A204027
GCD(F(i+1),F(j+1),F=A000045..A204112...A204113
GCD(L(i),L(j),L=A000032......A204114...A204115
GCD(2^i-1,2^j-2).............A204116...A204117
GCD(prime(i),prime(j)).......A204118...A204119
GCD(prime(i+1),prime(j+1))...A204120...A204121
GCD(2^(i-1),2^(j-1)).........A144464...A204122
max(floor(i/j),floor(j/i))...A204123...A204124
min(ceil(i/j),ceil(j/i)).....A204143...A204144
Delannoy matrix..............A008288...A204135
max(2i-j,2j-i)...............A204154...A204155
-1+max(3i-j,3j-i)............A204156...A204157
max(3i-2j,3j-2i).............A204158...A204159
floor[(i+1)/2]...............A204164...A204165
ceiling[(i+1)/2].............A204166...A204167
i+j..........................A003057...A204168
i+j-1........................A002024...A204169
i*j..........................A003991...A204170
..abbreviation below: AOE means "all 1's except"
AOE f(i,i)=i.................A204125...A204126
AOE f(i,i)=A000045(i+1)......A204127...A204128
AOE f(i,i)=A000032(i)........A204129...A204130
AOE f(i,i)=2i-1..............A204131...A204132
AOE f(i,i)=2^(i-1)...........A204133...A204134
AOE f(i,i)=3i-2..............A204160...A204161
AOE f(i,i)=floor[(i+1)/2]....A204162...A204163
...
Other pairs (M, c.p.s.): (A204171, A204172) to (A204183, A204184)
See A202695 for a guide to choices of symmetric matrix M for which the zeros of the characteristic polynomials are all positive.
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LINKS
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Table of n, a(n) for n=1..99.
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EXAMPLE
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Northwest corner:
0 1 1 1 1 1 1 1
0 1 2 2 2 2 2 2
1 2 0 3 3 3 3 3
1 2 3 0 4 4 4 4
1 2 3 4 0 5 5 5
1 2 3 4 5 0 6 6
1 2 3 4 5 6 0 7
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MATHEMATICA
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f[i_, j_] := Max[Mod[i, j], Mod[j, i]];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
{n, 1, 12}, {i, 1, n}]] (* A204016 *)
p[n_] := CharacteristicPolynomial[m[n], x];
c[n_] := CoefficientList[p[n], x]
TableForm[Flatten[Table[p[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%] (* A204017 *)
TableForm[Table[c[n], {n, 1, 10}]]
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CROSSREFS
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Cf. A204017, A202453.
Sequence in context: A208183 A214810 A090737 * A157865 A072550 A037810
Adjacent sequences: A204013 A204014 A204015 * A204017 A204018 A204019
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KEYWORD
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nonn,tabl
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AUTHOR
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Clark Kimberling, Jan 10 2012
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STATUS
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approved
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