

A092297


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


12



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

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 nth power of the adjacency matrix of the complete 3graph, a 3 X 3 matrix with diagonal elements all zero and offdiagonal 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



FORMULA

a(n) = 2^n + 2*(1)^n; recurrence a(1)=0, a(2)=6, a(n) = 2*a(n2) + a(n1).
O.g.f: 6*x^2/((1+x)*(2*x1)) = 3  1/(2*x1) + 2/(1+x).  R. J. Mathar, Dec 02 2007
a(n) = (1)^n * a(2n) * 2^(n1) 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/(13x/(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

(PARI) {a(n) = 2 * (2^(n1)  (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



