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A204017
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Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of max{j mod i, i mod j} (A204016).
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2
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0, -1, -1, 0, 1, 4, 6, 0, -1, -15, -38, -20, 0, 1, 56, 206, 184, 50, 0, -1, -185, -1072, -1357, -630, -105, 0, 1, 204, 5146, 9276, 6060, 1736, 196, 0, -1, 6209, -17334, -58470, -52452, -21102, -4116, -336, 0, 1, -112400, -67682, 293984
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OFFSET
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1,6
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COMMENTS
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Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are real, and they interlace the zeros of p(n+1). See A204016 and A202605 for guides to related sequences.
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REFERENCES
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(For references regarding interlacing roots, see A202605.)
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LINKS
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EXAMPLE
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Top of the array:
1... -1
-1.... 0.... 1
4.... 6.... 0... -1
-15.. -38.. -20... 0... 1
56... 206.. 184.. 50.. 0.. -1
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The 1st principal submatrix (ps) of A204016 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
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The 2nd ps is {{0,1},{1,0}}, with p(2)=-1+x^2 and zero-set {-1,1}.
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The 3rd ps is {{0,1,1},{1,0,2},{1,2,0}}, with p(3)=4+6x-x^3 and zero-set {-2, -0.732...,2.732...}.
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The 4th ps is {{0,1,1,1},{1,0,2,2},{1,2,0,3},{1,2,0,3}}, with p(4)=-15-38x-20x^2+x^4 and zero-set {-3, -1.714, -0.553, 5.268}.
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The interlace property is illustrated for the last two zero-sets by this chain:
-3 < -2 < -1.7 < -0.7 < -0.5 < 2.7 < 5.2
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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}]
TableForm[Table[c[n], {n, 1, 10}]]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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