A263318 Number of aperiodic necklaces (Lyndon words) with 9 black beads and n white beads.

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

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

Index entries for linear recurrences with constant coefficients, signature (6, -15, 23, -33, 51, -64, 63, -63, 64, -51, 33, -23, 15, -6, 1).

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 *)
LinearRecurrence[{6, -15, 23, -33, 51, -64, 63, -63, 64, -51, 33, -23, 15, -6, 1}, {0, 1, 5, 18, 55, 143, 333, 715, 1430, 2700, 4862, 8398, 13995, 22610, 35530}, 40] (* Harvey P. Dale, Feb 10 2023 *)

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 A360191

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