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A335862
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Decimal expansion of the zero x1 of the cubic polynomial x^3 - 2*x^2 - 10*x - 6.
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2
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4, 5, 1, 1, 4, 0, 4, 6, 6, 4, 2, 2, 6, 7, 5, 8, 1, 2, 3, 3, 3, 9, 2, 2, 1, 4, 9, 6, 8, 1, 3, 1, 6, 9, 5, 7, 4, 0, 2, 1, 8, 4, 3, 6, 1, 6, 4, 5, 0, 8, 8, 5, 7, 4, 6, 3, 5, 1, 7, 4, 8, 6, 8, 6, 1, 2, 7, 9, 5, 8, 3, 4, 4, 8, 2, 1, 6, 4, 9, 2, 5, 1, 5, 8, 9, 6, 7, 5, 8, 2, 7, 1, 7, 4, 3, 2, 5, 5, 3, 3
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OFFSET
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1,1
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COMMENTS
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This cubic polynomial P3(x) = x^3 - 2*x^2 - 10*x - 6 is a factor of the characteristic polynomial F(x) of degree 7 of the 7 X 7 adjacency matrix F7 of the Fano graph with nodes (vertices) of degree 6, 5, 5, 5, 3, 3, 3. See the links for the Fano plane. The graph is in fact planar.
The adjacency matrix is F7 = Matrix([[0, 1, 1, 1, 1, 1, 1], [1, 0, 1, 1, 1, 1, 0], [1, 1, 0, 1, 0, 1, 1], [1, 1, 1, 0, 1, 0, 1], [1, 1, 0, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0, 0], [1, 0, 1, 1, 0, 0, 0]]).
The determinant of F7 is 6. The characteristic polynomial is F(x) = x^7 - 15*x^5 - 26*x^4 + 3*x^3 + 24*x^2 + 2*x - 6 = P3(x)*(x^2 + x - 1)^2. The zeros of F(x) (the eigenvalues or spectrum of F7) are: x1, x2 = -A335863 = -1.752517821..., x3 = -A335864 = -0.7588868422..., twice -1 + phi = 0.618033988..., and twice -phi, where phi = A001622.
For the bipartite incidence graph see the links for the Heawood graph.
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LINKS
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FORMULA
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x1 = (1/3)*(2 + (179 + 3*sqrt(3)*sqrt(269)*i)^(1/3) + ( 179 - 3*sqrt(3)*sqrt(269)*i)^(1/3)), where i is the imaginary unit.
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EXAMPLE
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x1 = 4.5114046642267581233392214968131695740218436164...
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MATHEMATICA
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With[{k = 3 Sqrt[3] Sqrt[269] I}, First@ RealDigits[Re[(1/3) (2 + (179 + k)^(1/3) + (179 - k)^(1/3))], 10, 100]] (* Michael De Vlieger, Nov 17 2020 *)
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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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