OFFSET
6,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 6..1000
Index entries for linear recurrences with constant coefficients, signature (15,-90,270,-405,246,-45,270,-810,1215,-729).
FORMULA
O.g.f.: 3*x^6*(1 - 6*x + 18*x^2 - 27*x^3 + 16*x^4)/((1 - 3*x)^5*(1 - 3*x^5)).
O.g.f.: (1/5)*((x/(1-3*x))^5 - x^5/(1-3*x^5)).
a(n) = (1/5)*Sum_{d|5, d|n} mu(d) C(n/d-1, (n-5)/d )*3^((n-5)/d).
a(n) = (1/5)*C(n-1, 4)*3^(n-5) if n=1,2,3,4 mod 5.
a(n) = (1/5)*C(n-1, 4)*3^(n-5) - (1/5)*3^((n-5)/5) if n=0 mod 5.
EXAMPLE
a(7) = 27 because 11111ab, 1111a1b, 111a11b for a,b=2,3,4 are all Lyndon of length 7
MATHEMATICA
3*(1 -6*x +18*x^2 -27*x^3 +16*x^4)/((1-3*x)^5*(1-3*x^5)) + O[x]^22 // CoefficientList[#, x]& (* Jean-François Alcover, Sep 19 2017 *)
PROG
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 3*x^6*(1-6*x+18*x^2-27*x^3+16*x^4)/((1-3*x)^5*(1-3*x^5)) )); // G. C. Greubel, Aug 17 2023
(SageMath)
def f(x): return 3*x^6*(1-6*x+18*x^2-27*x^3+16*x^4)/((1-3*x)^5*(1-3*x^5)
def A124813_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( f(x) ).list()
a=A124813_list(46); a[6:] # G. C. Greubel, Aug 17 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Mike Zabrocki, Nov 08 2006
STATUS
approved