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A290743 Maximum number of distinct Lyndon factors that can appear in words of length n over an alphabet of size 2. 5
2, 3, 4, 6, 8, 11, 14, 18, 22, 27, 32, 38, 44, 51, 58, 66, 74, 83, 92, 102, 112, 123, 134, 146, 158, 171, 184, 198, 212, 227, 242, 258, 274, 291, 308, 326, 344, 363, 382, 402, 422, 443, 464, 486, 508, 531, 554, 578, 602, 627, 652, 678, 704, 731, 758 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

See theorem 1 of reference for formula.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

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

FORMULA

a(n) = binomial(n+1,2) - (s-p)*binomial(m+1,2) - p*binomial(m+2,2) + s where s=2, m=floor(n/s), p=n-m*s. - Andrew Howroyd, Aug 14 2017

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

G.f.: x*(2 - x - 2*x^2 + 2*x^3) / ((1 - x)^3*(1 + x)).

a(n) = (2*n^2 + 16) / 8 for n even.

a(n) = (2*n^2 + 14) / 8 for n odd.

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 4.

(End)

MATHEMATICA

Table[(Binomial[n+1, 2] - (2-(n - 2 Floor[n/2])) Binomial[Floor[n/2]+1, 2] - (n-2 Floor[n/2]) Binomial[Floor[n/2]+2, 2] + 2), {n, 60}] (* Vincenzo Librandi, Oct 04 2017 *)

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)(2); \\ Andrew Howroyd, Aug 14 2017

(MAGMA) [Binomial(n+1, 2)-(2-(n-2*Floor(n/2)))*Binomial(Floor(n/2)+1, 2)-(n-2*Floor(n/2))*Binomial(Floor(n/2)+2, 2)+2: n in [1..60]]; // Vincenzo Librandi, Oct 04 2017

CROSSREFS

Cf. A290744, A290745, A290746.

Sequence in context: A211536 A071764 A238381 * A059291 A177339 A075535

Adjacent sequences:  A290740 A290741 A290742 * A290744 A290745 A290746

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Aug 11 2017

EXTENSIONS

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

STATUS

approved

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Last modified November 21 15:02 EST 2019. Contains 329371 sequences. (Running on oeis4.)