OFFSET
4,1
COMMENTS
The number of ways of m-coloring an annulus consisting of n zones joined like a pearl necklace is (m-1)^n + (-1)^n*(m-1), where m >= 3 (cf. A092297 for m=3). Now we must also color the central region.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 4..3000
S. B. Step, More information.
Index entries for linear recurrences with constant coefficients, signature (1,2).
FORMULA
m=4, a(n)=m*((m-2)^(n-1)+(-1)^(n-1)*(m-2)); recurrence m=4, b(1)=0, b(2)=(m-1)*(m-2), b(n)=(m-2)*b(n-2)+(m-3)*b(n-1), a(n)=m*b(n-1).
O.g.f.: -24*x^3 - 12*x + 6 - 8/(1+x) - 2/(2*x-1). - R. J. Mathar, Dec 02 2007
a(n) = 24*A001045(n-2). - R. J. Mathar, Aug 30 2008
a(n) = 2^(n+1) - 8*(-1)^n. - Vincenzo Librandi, Oct 10 2011
EXAMPLE
We can choose 4 colors to color the inside zone, therefore b(3)=6 because we can color one zone in the annulus in 3 colors, another in 2, the other in 1, so 3*2*1=6 in all and a(4)=4*6=24. We can also add a(3)=4*3*2=24 to this sequence.
MATHEMATICA
LinearRecurrence[{1, 2}, {24, 72}, 30] (* Harvey P. Dale, Jan 25 2020 *)
PROG
(Magma) [2^(n+1)-8*(-1)^n: n in [4..35]]; // Vincenzo Librandi, Oct 10 2011
CROSSREFS
KEYWORD
nonn
AUTHOR
S.B.Step (stepy(AT)vesta.ocn.ne.jp), Feb 12 2004
STATUS
approved