A124810 Number of 4-ary Lyndon words of length n with exactly two 1s.
3, 12, 54, 198, 729, 2538, 8748, 29484, 98415, 324648, 1062882, 3454002, 11160261, 35871174, 114791256, 365893848, 1162261467, 3680484804, 11622614670, 36611206686, 115063885233, 360882096930, 1129718145924, 3530368940292
Offset: 3
Examples
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.
Links
- Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160. [From _R. J. Mathar_, Nov 08 2008]
- Index entries for linear recurrences with constant coefficients, signature (6, -6, -18, 27).
Programs
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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); # Alois P. Heinz, Aug 04 2008
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Mathematica
a[n_] := (1/2)*(n-1)*3^(n-2) - If[OddQ[n], 0, (1/2)*3^((n-2)/2)]; Array[a, 24, 3] (* Jean-François Alcover, Sep 19 2017 *)
Formula
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.
a(2n+1) = A230540(n)/2. - R. J. Mathar, Jul 20 2025
Comments