login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A056292 Number of n-bead necklace structures using a maximum of four different colored beads. 11
1, 2, 3, 7, 11, 39, 103, 367, 1235, 4439, 15935, 58509, 215251, 799697, 2983217, 11187567, 42109451, 159082753, 602809327, 2290684251, 8726308317, 33318661277, 127479700199, 488672302909, 1876500180291, 7217308815887, 27799998949873, 107228568948547 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Turning over the necklace is not allowed. Colors may be permuted without changing the necklace structure.

REFERENCES

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

LINKS

Table of n, a(n) for n=1..28.

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Marko Riedel, Necklaces with swappable colors by Power Group Enumeration

Marko Riedel, Maple code for any necklace size, any number of swappable colors, by Power Group Enumeration.

N. J. A. Sloane, Maple code for this and related sequences

FORMULA

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.

From Robert A. Russell, May 29 2018: (Start)

a(n) = (1/n) * Sum_{d|n} phi(d) * ([d==0 mod 12] * (4*S2(n/d+3, 4) - 24*S2(n/d+2, 4) + 44*S2(n/d+1, 4) - 24*S2(n/d, 4)) + [d==6 mod 12] * (3*S2(n/d+3, 4) - 18*S2(n/d+2, 4) + 33*S2(n/d+1, 4) - 18*S2(n/d, 4)) + [d==4 mod 12 | d==8 mod 12] * (3*S2(n/d+3, 4) - 19*S2(n/d+2, 4) + 38*S2(n/d+1, 4) - 24*S2(n/d, 4)) + [d==3 mod 12 | d=9 mod 12] * (2*S2(n/d+3, 4) - 13*S2(n/d+2, 4) + 26*S2(n/d+1, 4) - 15*S2(n/d, 4)) + [d==2 mod 12 | d=10 mod 12] * (2*S2(n/d+3, 4) - 13*S2(n/d+2, 4) + 27*S2[n/d+1,4) - 18*S2(n/d, 4)) + [d mod 12 in {1,5,7,11}] * (S2(n/d+3, 4) - 8*S2(n/d+2, 4) + 20*S2(n/d+1, 4) - 15*S2(n/d, 4))), where S2(n, k) is the Stirling subset number, A008277.

G.f.: 1 - Sum_{d>0} (phi(d) / d) * ([d==0 mod 12] * log(1-4x^d) + [d==6 mod 12] * 3*log(1-4x^d) / 4 + [d==4 mod 12 | d==8 mod 12] * (2*log(1-4x^d) + log(1-x^d)) / 3 + [d==3 mod 12 | d=9 mod 12] * (3*log(1-4x^d) + 2*log(1-2x^d)) / 8 + [d==2 mod 12 | d=10 mod 12] * (5*log(1-4x^d) + 4*log(1-x^d)) / 12 + [d mod 12 in {1,5,7,11}] * (log(1-4x^d) + 6*log(1-2x^d) + 8*log(1-x^d)) / 24).

(End)

MATHEMATICA

Adn[d_, n_] := Module[{ c, t1, t2}, t2 = 0; For[c = 1, c <= d, c++, If[Mod[d, c] == 0 , t2 = t2 + (x^c/c)*(E^(c*z) - 1)]]; t1 = E^t2; t1 = Series[t1, {z, 0, n+1}]; Coefficient[t1, z, n]*n!]; Pn[n_] := Module[{ d, e, t1}, t1 = 0; For[d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*Adn[d, n/d]/n]]; t1/(1 - x)]; Pnq[n_, q_] := Module[{t1}, t1 = Series[Pn[n], {x, 0, q+1}] ; Coefficient[t1, x, q]]; a[n_] := Pnq[n, 4]; Table[Print[an = a[n]]; an, {n, 1, 25}] (* Jean-Fran├žois Alcover, Oct 04 2013, after N. J. A. Sloane's Maple code *)

(* This program uses Gilbert and Riordan's recurrence formula, which they recommend for calculations: *)

Adn[d_, n_] := Adn[d, n] = If[1==n, DivisorSum[d, x^# &],

  Expand[Adn[d, 1] Adn[d, n-1] + D[Adn[d, n-1], x] x]];

Table[SeriesCoefficient[DivisorSum[n, EulerPhi[#] Adn[#, n/#] &]

/(n (1 - x)), {x, 0, 4}], {n, 1, 40}] (* Robert A. Russell, Feb 24 2018 *)

From Robert A. Russell, May 29 2018: (Start)

Table[(1/n) DivisorSum[n, EulerPhi[#] Which[Divisible[#, 12], 4 StirlingS2[n/#+3, 4] - 24 StirlingS2[n/#+2, 4] + 44 StirlingS2[n/#+1, 4] - 24 StirlingS2[n/#, 4], Divisible[#, 6], 3 StirlingS2[n/#+3, 4] - 18 StirlingS2[n/#+2, 4] + 33 StirlingS2[n/#+1, 4] - 18 StirlingS2[n/#, 4], Divisible[#, 4], 3 StirlingS2[n/#+3, 4] - 19 StirlingS2[n/#+2, 4] + 38 StirlingS2[n/#+1, 4] - 24 StirlingS2[n/#, 4], Divisible[#, 3], 2 StirlingS2[n/#+3, 4] - 13 StirlingS2[n/#+2, 4] + 26 StirlingS2[n/#+1, 4] - 15 StirlingS2[n/#, 4], Divisible[#, 2], 2 StirlingS2[n/#+3, 4] - 13 StirlingS2[n/#+2, 4] + 27 StirlingS2[n/#+1, 4] - 18 StirlingS2[n/#, 4], True, StirlingS2[n/#+3, 4] - 8 StirlingS2[n/#+2, 4] + 20 StirlingS2[n/#+1, 4] - 15 StirlingS2[n/#, 4]] &], {n, 1, 40}]

mx = 40; Drop[CoefficientList[Series[1 - Sum[(EulerPhi[d] / d) Which[

  Divisible[d, 12], Log[1 - 4x^d], Divisible[d, 6],

  3 Log[1 - 4x^d] / 4, Divisible[d, 4] ,

  (2 Log[1 - 4x^d] + Log[1 - x^d]) / 3, Divisible[d, 3],

  (3 Log[1 - 4x^d] + 2 Log[1 - 2x^d]) / 8,

  Divisible[d, 2], (5 Log[1 - 4x^d] + 4 Log[1 - x^d]) / 12,

  True, (Log[1 - 4x^d] + 6 Log[1 - 2x^d] + 8 Log[1 - x^d]) / 24], {d, 1, mx}], {x, 0, mx}], x], 1]

(End)

CROSSREFS

Cf. A000013, A001868.

Sequence in context: A034295 A056354 A072534 * A106125 A175171 A073609

Adjacent sequences:  A056289 A056290 A056291 * A056293 A056294 A056295

KEYWORD

nonn

AUTHOR

Marks R. Nester

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 30 22:55 EDT 2020. Contains 333132 sequences. (Running on oeis4.)