A261882 Decimal expansion of 32/27.

1, 1, 8, 5, 1, 8, 5, 1, 8, 5, 1, 8, 5, 1, 8, 5, 1, 8, 5, 1, 8, 5, 1, 8, 5, 1, 8, 5, 1, 8, 5, 1, 8, 5, 1, 8, 5, 1, 8, 5, 1, 8, 5, 1, 8, 5, 1, 8, 5, 1, 8, 5, 1, 8, 5, 1, 8, 5, 1, 8, 5, 1, 8, 5, 1, 8, 5

Offset: 1

Author

Charles R Greathouse IV, Sep 04 2015

Keywords

• nonn
• cons
• easy

Comments

For any number x >= 32/27 and any e > 0, there is a graph G such that the chromatic polynomial of G has a real root between x - e and x + e. (All real roots of such polynomials are 0, 1, or in this range.)

Continued fraction expansion of (sqrt(730)-10)/9. - Bruno Berselli, Sep 04 2015

Periodic (beyond the first term) with period 3. - Charles R Greathouse IV, Sep 05 2015

Equals the ratio of the wavelengths between the hydrogen spectral lines Lyman-alpha (121.6 nm) and Lyman-beta (102.6 nm). - Sean Stroud, Apr 15 2019

Examples

			1.18518518518518518...
		

Links

• Peter J. Cameron's Blog, Algebraic properties of chromatic roots, Oct 04 2016.
• Bill Jackson, A zero-free interval for chromatic polynomials of graphs, Combinatorics, Probability and Computing 2:3 (Sept 1993), pp. 325-336.
• Bill Jackson and Alan Sokal, Zero-free regions for multivariate Tutte polynomials (alias Potts-model partition functions) of graphs and matroids, J. Combin. Theory Ser. B 99:6 (2009), pp. 869-903.
• Carsten Thomassen, The zero-free intervals for chromatic polynomials of graphs, Combin. Probab. Comput. 6:4 (1997), pp. 497-506.
• Index entries for linear recurrences with constant coefficients, signature (0, 0, 1).

Crossrefs

Cf. A021058.

Programs

Maple

Digits := 100; evalf(32/27); # Wesley Ivan Hurt, Sep 04 2015

Mathematica

First@ RealDigits[N[32/27, 120]] (* Michael De Vlieger, Sep 04 2015 *)
Join[{1}, Table[7 - (-1)^Mod[n - 1, 3]/2 - 5 (-1)^Mod[n, 3]/2 - 4 (-1)^Mod[n + 1, 3], {n, 2, 40}]] (* Wesley Ivan Hurt, Sep 04 2015 *)

PARI

32/27.

Formula

G.f.: x*(1 + x + 8*x^2 + 4*x^3)/((1 - x)*(1 + x + x^2)). - Bruno Berselli, Sep 04 2015

a(n) = 7-(-1)^(n-1 mod 3)/2-5*(-1)^(n mod 3)/2-4*(-1)^(n+1 mod 3), n>1. - Wesley Ivan Hurt, Sep 04 2015