A290744 Maximum number of distinct Lyndon factors that can appear in words of length n over an alphabet of size 5.

5, 6, 8, 11, 15, 19, 24, 30, 37, 45, 53, 62, 72, 83, 95, 107, 120, 134, 149, 165, 181, 198, 216, 235, 255, 275, 296, 318, 341, 365, 389, 414, 440, 467, 495, 523, 552, 582, 613, 645, 677, 710, 744, 779, 815, 851, 888, 926, 965, 1005, 1045, 1086, 1128, 1171, 1215

Offset

1,1

Links

Table of n, a(n) for n=1..55.

Amy Glen, Jamie Simpson, and W. F. Smyth, Counting Lyndon Factors, Electronic Journal of Combinatorics 24(3) (2017), #P3.28.

Ryo Hirakawa, Yuto Nakashima, Shunsuke Inenaga, and Masayuki Takeda, Counting Lyndon Subsequences, arXiv:2106.01190 [math.CO], 2021. See MDF(n, s).

Formula

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

Conjectures from Colin Barker, Oct 03 2017: (Start)

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)).

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

(End)

Prog

(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

Crossrefs

Cf. A290743, A290745, A290747.

Sequence in context: A332549 A091091 A262161 * A120151 A022937 A030742

Adjacent sequences: A290741 A290742 A290743 * A290745 A290746 A290747

Keyword

nonn

Author

N. J. A. Sloane, Aug 11 2017

Extensions

a(11)-a(55) from Andrew Howroyd, Aug 14 2017

Status

approved