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 *)
LinearRecurrence[{1, 2}, {0, 6}, 40] (* Harvey P. Dale, May 21 2024 *)
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
KEYWORD
nonn,easy
AUTHOR
S. B. Step (stepy(AT)vesta.ocn.ne.jp), Feb 06 2004
STATUS
approved