A090860 Number of ways of 4-coloring a map in which there is a central circle surrounded by an annulus divided into n-1 regions. There are n regions in all.

24, 72, 120, 264, 504, 1032, 2040, 4104, 8184, 16392, 32760, 65544, 131064, 262152, 524280, 1048584, 2097144, 4194312, 8388600, 16777224, 33554424, 67108872, 134217720, 268435464, 536870904, 1073741832, 2147483640, 4294967304

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

Cf. A001045, A092297.

Sequence in context: A006352 A143337 A183006 * A304374 A064200 A305065

Adjacent sequences: A090857 A090858 A090859 * A090861 A090862 A090863

Keyword

nonn

Author

S.B.Step (stepy(AT)vesta.ocn.ne.jp), Feb 12 2004

Status

approved