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A204115
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Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix from A204114, given by GCD(L(i+1),L(j+1)), where L=A000032 (Lucas numbers).
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3
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1, -1, 2, -4, 1, 6, -16, 8, -1, 36, -108, 69, -15, 1, 360, -1152, 834, -230, 26, -1, 5280, -17696, 14368, -4668, 682, -44, 1, 147840, -506048, 426568, -147856, 24262, -1952, 73, -1, 6800640, -23573888, 20317360
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OFFSET
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1,3
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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 A202605 and A204016 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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Table of n, a(n) for n=1..38.
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EXAMPLE
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Top of the array:
1....-1
2....-4.....1
6....-16....8....-1
36...-108...69...-15....1
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MATHEMATICA
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u[n_] := LucasL[n]
f[i_, j_] := GCD[u[i], u[j]];
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, 15}, {i, 1, n}]] (* A204114 *)
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[%] (* A204115 *)
TableForm[Table[c[n], {n, 1, 10}]]
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CROSSREFS
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Cf. A204114, A202605, A204016.
Sequence in context: A167546 A011369 A110877 * A204130 A204024 A021009
Adjacent sequences: A204112 A204113 A204114 * A204116 A204117 A204118
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KEYWORD
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tabl,sign
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AUTHOR
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Clark Kimberling, Jan 11 2012
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STATUS
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approved
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