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A090860
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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.
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2
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 4,1
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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.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 4..3000
S. B. Step, More information
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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 (mathar(AT)strw.leidenuniv.nl), Dec 02 2007
a(n)=24*A001045(n-2). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 30 2008]
a(n) = 2^(n+1)-8*(-1)^n. - Vincenzo Librandi, Oct 10 2011
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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.
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PROG
| (MAGMA) [2^(n+1)-8*(-1)^n: n in [4..35]]; // Vincenzo Librandi, Oct 10 2011
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CROSSREFS
| Cf. A092297.
Sequence in context: A006352 A143337 A183006 * A064200 A192833 A189540
Adjacent sequences: A090857 A090858 A090859 * A090861 A090862 A090863
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KEYWORD
| nonn
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AUTHOR
| S.B.Step (stepy(AT)vesta.ocn.ne.jp), Feb 12 2004
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