%I #36 Apr 09 2024 14:21:00

%S 24,72,120,264,504,1032,2040,4104,8184,16392,32760,65544,131064,

%T 262152,524280,1048584,2097144,4194312,8388600,16777224,33554424,

%U 67108872,134217720,268435464,536870904,1073741832,2147483640,4294967304

%N 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.

%C 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.

%H Vincenzo Librandi, <a href="/A090860/b090860.txt">Table of n, a(n) for n = 4..3000</a>

%H S. B. Step, <a href="http://hp.vector.co.jp/authors/VA008683/english/index.htm">More information</a>.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,2).

%F 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).

%F O.g.f.: -24*x^3 - 12*x + 6 - 8/(1+x) - 2/(2*x-1). - _R. J. Mathar_, Dec 02 2007

%F a(n) = 24*A001045(n-2). - _R. J. Mathar_, Aug 30 2008

%F a(n) = 2^(n+1) - 8*(-1)^n. - _Vincenzo Librandi_, Oct 10 2011

%e 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.

%t LinearRecurrence[{1,2},{24,72},30] (* _Harvey P. Dale_, Jan 25 2020 *)

%o (Magma) [2^(n+1)-8*(-1)^n: n in [4..35]]; // _Vincenzo Librandi_, Oct 10 2011

%Y Cf. A001045, A092297.

%K nonn

%O 4,1

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