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A124811 Number of 4-ary Lyndon words of length n with exactly three 1s. 4

%I

%S 3,18,89,405,1701,6801,26244,98415,360846,1299078,4605822,16120350,

%T 55801305,191318760,650483703,2195382771,7360989291,24536630727,

%U 81358302690,268482398877,882156452724,2887057484028,9414317882700,30596533116588,99132767304831

%N Number of 4-ary Lyndon words of length n with exactly three 1s.

%F o.g.f. x^4 (3-9 x+8 x^2)/(1-3x)/(1-3x^3) = 1/3*((x/(1-3*x))^3 - x^3/(1-3*x^3)) a(n) = 1/3*sum_{d|3,d|n} mu(d) C(n/d-1,(n-3)/d )*3^((n-3)/d) = 1/3*C(n-1,2)*3^(n-3) if d=1,2 mod 3 = 1/3*C(n-1,2)*3^(n-3) - 1/3*3^((n-3)/3) if d=0 mod 3

%e a(5) = 18 because 111ab and 11a1b are Lyndon of length 4 for ab=2,3,4.

%t (3 - 9*x + 8*x^2)/((1 - 3*x)^3*(1 - 3*x^3)) + O[x]^23 // CoefficientList[#, x]& (* _Jean-Fran├žois Alcover_, Sep 19 2017 *)

%Y Cf. A124810, A124812, A124813, A124814, A001840, A124721.

%K nonn

%O 4,1

%A _Mike Zabrocki_, Nov 08 2006

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Last modified September 28 09:32 EDT 2021. Contains 347714 sequences. (Running on oeis4.)