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Search: id:A124810
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| A124810 |
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Number of 4-ary Lyndon words of length n with exactly two 1s. |
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+0
5
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| 3, 12, 54, 198, 729, 2538, 8748, 29484, 98415, 324648, 1062882, 3454002, 11160261, 35871174, 114791256, 365893848, 1162261467, 3680484804, 11622614670, 36611206686, 115063885233, 360882096930, 1129718145924, 3530368940292
(list; graph; listen)
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OFFSET
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3,1
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COMMENT
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If the offsets are modified, A124810 to A124813 are the 2nd to 5th Witt transform of A000244 [Moree]. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 08 2008]
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LINKS
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Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 08 2008]
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FORMULA
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o.g.f. 3 x^3 (1-2 x)/((1-3x)^2 (1-3x^2)) = 1/2*((x/(1-3*x))^2 - x^2/(1-3*x^2)) a(n) = 1/2*sum_{d|2,d|n} mu(d) C(n/d-1,(n-2)/d )*3^((n-2)/d) =1/2*(n-1)*3^(n-2) if n is odd =1/2*(n-1)*3^(n-2) - 1/2*3^((n-2)/2) if n is even
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EXAMPLE
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a(4) = 12 because 1122, 1123, 1124, 1132, 1133, 1134, 1142, 1143, 1144, 1213, 1214, 1314 are all 4-ary Lyndon words with length 4 and have exactly two 1s.
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MAPLE
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(Maple) a := n -> (Matrix([[12, 3, 0, 0]]). Matrix(4, (i, j)-> if (i=j-1) then 1 elif j=1 then [6, -6, -18, 27][i] else 0 fi)^(n-4))[1, 1] ; seq (a(n), n=3..26); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 04 2008]
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CROSSREFS
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Cf. A124811, A124812, A124813, A124814, A004526, A124720.
Sequence in context: A110122 A060460 A120983 this_sequence A123348 A151204 A151205
Adjacent sequences: A124807 A124808 A124809 this_sequence A124811 A124812 A124813
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KEYWORD
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nonn
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AUTHOR
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Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Nov 08 2006
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