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
Keywords
Links
- Pedro Antonio, Table of n, a(n) for n = 0..100
- David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
- 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).
Programs
-
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 *) -
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
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
Extensions
More terms from Michel Marcus, Oct 14 2015
Comments