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A124812 Number of 4-ary Lyndon words of length n with exactly four 1s. 4
3, 21, 135, 702, 3402, 15282, 65610, 270540, 1082565, 4221639, 16120377, 60450138, 223205220, 813100356, 2927177028, 10428053400, 36804946455, 128817263385, 447470664795, 1543773631158, 5292938720718, 18044108743734, 61193066237550 (list; graph; refs; listen; history; text; internal format)
OFFSET

5,1

LINKS

Table of n, a(n) for n=5..27.

FORMULA

o.g.f. 3 x^5 (1-5 x + 9 x^2 - 6 x^3)/((1-3 x^2)^2 (1- 3 x)^4) = 1/4*((x/(1-3*x))^4 - x^4/(1-3*x^2)^2) a(n) = 1/4*sum_{d|4,d|n} mu(d) C(n/d-1,(n-4)/d )*3^((n-4)/d) = 1/4*C(n-1,3)*3^(n-4) if n is odd = 1/4*C(n-1,3)*3^(n-4) - 1/4*(n/2-1)*3^((n-4)/2) if n is even

EXAMPLE

a(6) = 21 because 1111ab, 1111ba, 111a1b, 111b1a, 11a11b for ab = 23, 24, 34 (accounting for 15 words) and 1111aa, 111a1a for a=2,3,4 (accounting for 6 words) are all Lyndon of length 6

MATHEMATICA

3*(1 - 5*x + 9*x^2 - 6*x^3)/((1 - 3*x)^4*(1 - 3*x^2)^2) + O[x]^23 // CoefficientList[#, x]& (* Jean-Fran├žois Alcover, Sep 19 2017 *)

CROSSREFS

Cf. A124810, A124811, A124813, A124814, A006918, A124722.

Sequence in context: A125701 A274586 A333030 * A141041 A079753 A137969

Adjacent sequences:  A124809 A124810 A124811 * A124813 A124814 A124815

KEYWORD

nonn

AUTHOR

Mike Zabrocki, Nov 08 2006

STATUS

approved

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Last modified June 4 01:24 EDT 2020. Contains 334808 sequences. (Running on oeis4.)