A290744 - OEIS

%I #19 Jun 04 2021 02:37:05

%S 5,6,8,11,15,19,24,30,37,45,53,62,72,83,95,107,120,134,149,165,181,

%T 198,216,235,255,275,296,318,341,365,389,414,440,467,495,523,552,582,

%U 613,645,677,710,744,779,815,851,888,926,965,1005,1045,1086,1128,1171,1215

%N Maximum number of distinct Lyndon factors that can appear in words of length n over an alphabet of size 5.

%H Amy Glen, Jamie Simpson, and W. F. Smyth, <a href="https://doi.org/10.37236/6915">Counting Lyndon Factors</a>, Electronic Journal of Combinatorics 24(3) (2017), #P3.28.

%H Ryo Hirakawa, Yuto Nakashima, Shunsuke Inenaga, and Masayuki Takeda, <a href="https://arxiv.org/abs/2106.01190">Counting Lyndon Subsequences</a>, arXiv:2106.01190 [math.CO], 2021. See MDF(n, s).

%F a(n) = binomial(n+1,2) - (s-p)*binomial(m+1,2) - p*binomial(m+2,2) + s where s=5, m=floor(n/s), p=n-m*s. - _Andrew Howroyd_, Aug 14 2017

%F Conjectures from _Colin Barker_, Oct 03 2017: (Start)

%F G.f.: x*(5 - 4*x + x^2 + x^3 + x^4 - 5*x^5 + 5*x^6) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)).

%F a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7) for n>6.

%F (End)

%o (PARI) a(n)=(s->my(m=n\s, p=n%s); binomial(n+1, 2)-(s-p)*binomial(m+1, 2)-p*binomial(m+2, 2)+s)(5); \\ _Andrew Howroyd_, Aug 14 2017

%Y Cf. A290743, A290745, A290747.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Aug 11 2017

%E a(11)-a(55) from _Andrew Howroyd_, Aug 14 2017