A335862 Decimal expansion of the zero x1 of the cubic polynomial x^3 - 2*x^2 - 10*x - 6.

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

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..100.

Wolfdieter Lang, A list of representative simple difference sets of the Singer type for small orders m, Karlsruher Institut für Technologie (Karlsruhe, Germany 2020).

Eric Weisstein's World of Mathematics, Fano plane

Eric Weisstein's World of Mathematics, Heawood graph

Wikipedia, Fano plane

Wikipedia, Heawood graph

Formula

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.

Example

x1 = 4.5114046642267581233392214968131695740218436164...

Mathematica

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 *)

Crossrefs

Cf. A001622, A335863 (-x2), A335864 (-x3).

Sequence in context: A121906 A028360 A010301 * A200385 A046576 A374035

Adjacent sequences: A335859 A335860 A335861 * A335863 A335864 A335865

Keyword

nonn,cons,easy

Author

Wolfdieter Lang, Jun 29 2020

Status

approved