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A263318 Number of aperiodic necklaces (Lyndon words) with 9 black beads and n white beads. 1
0, 1, 5, 18, 55, 143, 333, 715, 1430, 2700, 4862, 8398, 13995, 22610, 35530, 54477, 81719, 120175, 173583, 246675, 345345, 476901, 650325, 876525, 1168695, 1542684, 2017356, 2615085, 3362260, 4289780, 5433714, 6835972, 8544965, 10616463, 13114465, 16112057 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A row of triangle A051168.

LINKS

Pedro Antonio, Table of n, a(n) for n = 0..100

Index entries for sequences related to Lyndon words

FORMULA

a(n) = (1/(n+9))*Sum_{d divides gcd(n+9,9)} mu(d)*binomial((n+9)/d, 9/d).

G.f.: (x*(x^4-x^3+3*x^2-x+1))/((x^2+x+1)^3*(1-x)^9).

G.f.: ((-1+x^3)^-3-(-1+x)^-9)/9. - Herbert Kociemba, Oct 16 2016

MATHEMATICA

CoefficientList[Series[(x (x^4 - x^3 + 3*x^2 - x + 1))/((x^2 + x + 1)^3 (1 - x)^9), {x, 0, 40}], x] (* Wesley Ivan Hurt, Oct 15 2015 *)

CoefficientList[Series[((-1+x^3)^-3-(-1+x)^-9)/9, {x, 0, 40}], x] (* Herbert Kociemba, Oct 16 2016 *)

PROG

(PARI) a(n)= (1/(n+9))*sumdiv(gcd(n+9, 9), d, moebius(d)*binomial( (n+9)/d , 9/d )); \\ Michel Marcus, Oct 14 2015

(Python)

from sympy import mobius, binomial, gcd, divisors

print([sum(mobius(d) * binomial((n + 9)//d, 9//d) for d in divisors(gcd(n + 9, 9))) // (n + 9) for n in range(51)]) # Indranil Ghosh, Mar 26 2017

CROSSREFS

Cf. A001840, A006918, A011795, A011796, A011797, A051168.

Sequence in context: A270990 A272558 A081492 * A011845 A099450 A145129

Adjacent sequences:  A263315 A263316 A263317 * A263319 A263320 A263321

KEYWORD

nonn

AUTHOR

Criel Merino, Pedro Antonio, Oct 14 2015

EXTENSIONS

More terms from Michel Marcus, Oct 14 2015

STATUS

approved

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Last modified June 7 01:26 EDT 2020. Contains 334836 sequences. (Running on oeis4.)