A124812 Number of 4-ary Lyndon words of length n with exactly four 1s.

3, 21, 135, 702, 3402, 15282, 65610, 270540, 1082565, 4221639, 16120377, 60450138, 223205220, 813100356, 2927177028, 10428053400, 36804946455, 128817263385, 447470664795, 1543773631158, 5292938720718, 18044108743734, 61193066237550

Offset: 5

Author

Mike Zabrocki, Nov 08 2006

Keywords

• nonn

Examples

			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
		

Links

• G. C. Greubel, Table of n, a(n) for n = 5..1005
• Index entries for linear recurrences with constant coefficients, signature (12,-48,36,234,-540,0,972,-729).

Crossrefs

Cf. A006918, A124722, A124810, A124811, A124813, A124814, A254006.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 3*(1-5*x+9*x^2-6*x^3)/((1-3*x)^4*(1-3*x^2)^2) )); // G. C. Greubel, Aug 09 2023

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-François Alcover, Sep 19 2017 *)
LinearRecurrence[{12,-48,36,234,-540,0,972,-729}, {3,21,135,702,3402, 15282,65610,270540}, 41] (* G. C. Greubel, Aug 09 2023 *)

SageMath

def A124812(n): return (3/4)*(3^(n-5)*binomial(n-1,3) - ((n-2)//2)*3^((n-6)//2)*((n-5)%2))
[A124812(n) for n in range(5,41)] # G. C. Greubel, Aug 09 2023

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).

G.f.: (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).

a(n) = (1/4)*C(n-1, 3)*3^(n-4) if n is odd, a(n) = (1/4)*( C(n-1, 3)*3^(n-4) - (n/2-1)*3^((n-4)/2) ) if n is even.

a(n) = (3/4)*( 3^(n-5)*binomial(n-1, 3) - ((n-2)/2)*A254006(n-6) ). - G. C. Greubel, Aug 09 2023