A092297 Number of ways of 3-coloring an annulus consisting of n zones joined like a pearl necklace.

0, 6, 6, 18, 30, 66, 126, 258, 510, 1026, 2046, 4098, 8190, 16386, 32766, 65538, 131070, 262146, 524286, 1048578, 2097150, 4194306, 8388606, 16777218, 33554430, 67108866, 134217726, 268435458, 536870910, 1073741826, 2147483646

Offset

1,2

Comments

A circular domain means a domain between two concentric circles and it is divided into n parts by n boundary lines perpendicular to the circles. Both sides of a line must have different colors. How many ways of coloring are there?

a(n) is also the multiple of six that's nearest to 2^n. - David Eppstein, Aug 31 2010

a(n) apparently is the trace of the n-th power of the adjacency matrix of the complete 3-graph, a 3 X 3 matrix with diagonal elements all zero and off-diagonal all ones (cf. A001045). If so, a(n) is the number of closed walks on the graph of length n. - Tom Copeland, Nov 06 2012

For n >= 2, a(n) is the number of length n words on 3 letters with no two consecutive like letters including the first and the last. Cf. A218034. - Geoffrey Critzer, Apr 05 2014

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

K. Böhmová, C. Dalfó, and C. Huemer, On cyclic Kautz digraphs, Preprint 2016.

Cristina Dalfó, From subKautz digraphs to cyclic Kautz digraphs, arXiv:1709.01882 [math.CO], 2017.

C. Dalfó, The spectra of subKautz and cyclic Kautz digraphs, Linear Algebra and its Applications, 531 (2017), p. 210-219.

P. P. Martin and S. F. Zakaria, Zeros of the 4-state Potts model partition function for the square lattice revisited, J. Stat. Mech. 084003 (2019). eq. (7).

Carlos I. Perez-Sanchez, The Spectral Action on quivers, arXiv:2401.03705 [math.RT], 2024.

Index entries for linear recurrences with constant coefficients, signature (1,2).

Formula

a(n) = 2^n + 2*(-1)^n; recurrence a(1)=0, a(2)=6, a(n) = 2*a(n-2) + a(n-1).

O.g.f: -6*x^2/((1+x)*(2*x-1)) = -3 - 1/(2*x-1) + 2/(1+x). - R. J. Mathar, Dec 02 2007

a(n) = 6*A001045(n-1). - R. J. Mathar, Aug 30 2008

a(n) = (-1)^n * a(2-n) * 2^(n-1) for all n in Z. - Michael Somos, Oct 25 2014

Example

a(2)=6 because we can color one zone in 3 colors and the other in 2, so 2*3=6 in all.

Mathematica

nn=28; Drop[CoefficientList[Series[6x^2/(1+x)^2/(1-3x/(1+x)), {x, 0, nn}], x], 1] (* Geoffrey Critzer, Apr 05 2014 *)
a[ n_] := 2 (2^(n - 1) + (-1)^n); (* Michael Somos, Oct 25 2014 *)
a[ n_] := If[ n < 1, -(-2)^(n - 1) a[2 - n] , (-1)^n HypergeometricPFQ[ Table[ -2, {k, n}], Table[ 1, {k, n - 1}], 1]] (* Michael Somos, Oct 25 2014 *)

Prog

(Magma) [2^n+2*(-1)^n : n in [1..40]]; // Vincenzo Librandi, Sep 27 2011
(PARI) {a(n) = 2 * (2^(n-1) - (-1)^n)}; /* Michael Somos, Oct 25 2014 */

Crossrefs

Column k=3 of A106512.

Cf. A001045.

Sequence in context: A328528 A161787 A342285 * A294669 A224711 A073096

Adjacent sequences: A092294 A092295 A092296 * A092298 A092299 A092300

Keyword

nonn,easy

Author

S. B. Step (stepy(AT)vesta.ocn.ne.jp), Feb 06 2004

Status

approved