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A092297
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Number of ways of 3-coloring an annulus consisting of n zones joined like a pearl necklace.
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11
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..1000
K. Böhmová, C. Dalfó, 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, S. F. Zakaria, Zeros of the 4-state Potts model partition function for the square lattice revisited, J. Stat. Mech. 084003 (2019). eq. (7).
Index entries for linear recurrences with constant coefficients, signature (1,2).
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FORMULA
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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
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EXAMPLE
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a(2)=6 because we can color one zone in 3 colors and the other in 2, so 2*3=6 in all.
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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Column k=3 of A106512.
Cf. A001045.
Sequence in context: A328528 A161787 A342285 * A294669 A224711 A073096
Adjacent sequences: A092294 A092295 A092296 * A092298 A092299 A092300
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KEYWORD
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nonn,easy
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AUTHOR
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S. B. Step (stepy(AT)vesta.ocn.ne.jp), Feb 06 2004
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STATUS
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approved
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