A056299 Number of n-bead necklace structures using exactly six different colored beads.

0, 0, 0, 0, 0, 1, 3, 36, 296, 2303, 16317, 110462, 717024, 4532105, 28046285, 170938814, 1029749994, 6149327905, 36477979041, 215304158916, 1265984738264, 7422971231829, 43433472086235, 253759842223290

Offset

1,7

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..24.

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

Formula

a(n) = A056294(n) - A056293(n).

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

a(n) = (1/n) * Sum_{d|n} phi(d) * ([d==0 mod 60] * (S2(n/d+5,6) -

10*S2(n/d+4,6) + 35*S2(n/d+3,6) - 50*S2(n/d+2,6) + 24*S2(n/d+1,6)) +

[d==30 mod 60] * (S2(n/d+5,6) - 12*S2(n/d+4,6) + 56*S2(n/d+3,6) -

123*S2(n/d+2,6) + 108*S2(n/d+1,6)) + [d==20 mod 60 | d==40 mod 60] *

(4*S2(n/d+4,6) - 44*S2(n/d+3,6) + 176*S2(n/d+2,6) - 296*S2(n/d+1,6) +

160*S2(n/d,6)) + [d==15 mod 60 | d==45 mod 60] * (3*S2(n/d+4,6) -

36*S2(n/d+3,6) + 159*S2(n/d+2,6) - 306*S2(n/d+1,6) + 225*S2(n/d,6)) +

[d mod 60 in {12,24,36,48}] * (S2(n/d+5,6) - 12*S2(n/d+4,6) +

59*S2(n/d+3,6) - 156*S2(n/d+2,6) + 228*S2(n/d+1,6) - 144*S2(n/d,6)) +

[d=10 mod 60 | d==50 mod 60] * (2*S2(n/d+4,6) - 23*S2(n/d+3,6) +

103*S2(n/d+2,6) - 212*S2(n/d+1,6) + 160*S2(n/d,6)) + [d mod 60 in

{6,18,42,54}] * (S2(n/d+5,6) - 14*S2(n/d+4,6) + 80*S2(n/d+3,6) -

229*S2(n/d+2,6) + 312*S2(n/d+1,6) - 144*S2(n/d,6)) + [d mod 60 in

{5,25,35,55}] * (2*S2(n/d+4,6) - 24*S2(n/d+3,6) + 106*S2(n/d+2,6) -

204*S2(n/d+1,6) + 145*S2(n/d,6)) + [d mod 60 in {4,8,16,28,32,44,52,56}] *

(2*S2(n/d+4,6) - 20*S2(n/d+3,6) + 70*S2(n/d+2,6) - 92*S2(n/d+1,6) +

16*S2(n/d,6)) + [d mod 60 in {3,9,21,27,33,39,51,57}] * (S2(n/d+4,6) -

12*S2(n/d+3,6) + 53*S2(n/d+2,6) - 102*S2(n/d+1,6) + 81*S2(n/d,6)) +

[d mod 60 in {2,14,22,26,34,38,46,58}] * (S2(n/d+3,6) - 3*S2(n/d+2,6) -

8*S2(n/d+1,6) + 16*S2(n/d,6)) + [d mod 60 in {1,7,11,13,17,19,23,29,31,37,

41,43,47,49,53,59}] * S2(n/d,6)), where S2(n,k) is the Stirling subset

number, A008277.

G.f.: -Sum_{d>0} (phi(d) / d) * ([d==0 mod 60] * (log(1-6x^d) -

log(1-5x^d)) + [d==30 mod 60] * (3*log(1-6x^d) - 3*log(1-5x^d) +

log(1-2x^d) - log(1-x^d)) / 4 + [d==20 mod 60 | d==40 mod 60] *

(5*log(1-6x^d) - 6*log(1-5x^d) + 2*log(1-3x^d) - 3*log(1-2x^d)) / 9 +

[d==15 mod 60 | d==45 mod 60] * (5*log(1-6x^d) - 6*log(1-5x^d) +

3*log(1-4x^d) - 4*log(1-3x^d) + 3*log(1-2x^d) - 6*log(1-x^d)) / 16 +

[d mod 60 in {12,24,36,48}] * (4*log(1-6x^d) - 4*log(1-5x^d) +

log(1-x^d)) / 5 + [d=10 mod 60 | d==50 mod 60] * (11*log(1-6x^d) -

15*log(1-5x^d) + 8*log(1-3x^d) - 3*log(1-2x^d) - 9*log(1-x^d)) / 36 +

[d mod 60 in {6,18,42,54}] * (11*log(1-6x^d) - 11*log(1-5x^d) +

5*log(1-2x^d) - log(1-x^d)) / 20 + [d mod 60 in {5,25,35,55}] *

(29*log(1-6x^d) - 30*log(1-5x^d) + 3*log(1-4x^d) - 4*log(1-3x^d) +

3*log(1-2x^d) - 30*log(1-x^d)) / 144 + [d mod 60 in {4,8,16,28,32,44,52,

56}] * (16*log(1-6x^d) - 21*log(1-5x^d) + 10*log(1-3x^d) -

15*log(1-2x^d) + 9*log(1-x^d)) / 45 + [d mod 60 in {3,9,21,27,33,39,51, 57}] * (9*log(1-6x^d) - 14*log(1-5x^d) + 15*log(1-4x^d) - 20*log(1-3x^d) +

15*log(1-2x^d) - 14*log(1-x^d)) / 80 + [d mod 60 in {2,14,22,26,34,38,46,

58}] * (19*log(1-6x^d) - 39*log(1-5x^d) + 40*log(1-3x^d) -

15*log(1-2x^d) - 9*log(1-x^d)) / 180 + [d mod 60 in {1,7,11,13,17,19,23, 29,31,37,41,43,47,49,53,59}] * (log(1-6x^d) - 6 log(1-5x^d) +

15 log(1-4x^d) - 20 log(1-3x^d) + 15 log(1-2x^d) - 6 log(1-x^d)) / 720).

(End)

Mathematica

From Robert A. Russell, May 29 2018: (Start)
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[Coefficient[DivisorSum[n, EulerPhi[#] Adn[#, n/#] &]/n , x, 6],
  {n, 1, 40}] (* after Gilbert and Riordan *)
Table[(1/n) DivisorSum[n, EulerPhi[#] Which[Divisible[#, 60], StirlingS2[n/#+5, 6] - 10 StirlingS2[n/#+4, 6] + 35 StirlingS2[n/#+3, 6] - 50 StirlingS2[n/#+2, 6] + 24 StirlingS2[n/#+1, 6], Divisible[#, 30], StirlingS2[n/#+5, 6] - 12 StirlingS2[n/#+4, 6] + 56 StirlingS2[n/#+3, 6] - 123 StirlingS2[n/#+2, 6] + 108 StirlingS2[n/#+1, 6], Divisible[#, 20], 4 StirlingS2[n/#+4, 6] - 44 StirlingS2[n/#+3, 6] + 176 StirlingS2[n/#+2, 6] - 296 StirlingS2[n/#+1, 6] + 160 StirlingS2[n/#, 6], Divisible[#, 15], 3 StirlingS2[n/#+4, 6] - 36 StirlingS2[n/#+3, 6] + 159 StirlingS2[n/#+2, 6] - 306 StirlingS2[n/#+1, 6] + 225 StirlingS2[n/#, 6], Divisible[#, 12], StirlingS2[n/#+5, 6] - 12 StirlingS2[n/#+4, 6] + 59 StirlingS2[n/#+3, 6] - 156 StirlingS2[n/#+2, 6] + 228 StirlingS2[n/#+1, 6] - 144 StirlingS2[n/#, 6], Divisible[#, 10], 2 StirlingS2[n/#+4, 6] - 23 StirlingS2[n/#+3, 6] + 103 StirlingS2[n/#+2, 6] - 212 StirlingS2[n/#+1, 6] + 160 StirlingS2[n/#, 6], Divisible[#, 6], StirlingS2[n/#+5, 6] - 14 StirlingS2[n/#+4, 6] + 80 StirlingS2[n/#+3, 6] - 229 StirlingS2[n/#+2, 6] + 312 StirlingS2[n/#+1, 6] - 144 StirlingS2[n/#, 6], Divisible[#, 5], 2 StirlingS2[n/#+4, 6] - 24 StirlingS2[n/#+3, 6] + 106 StirlingS2[n/#+2, 6] - 204 StirlingS2[n/#+1, 6] + 145 StirlingS2[n/#, 6], Divisible[#, 4], 2 StirlingS2[n/#+4, 6] - 20 StirlingS2[n/#+3, 6] + 70 StirlingS2[n/#+2, 6] - 92 StirlingS2[n/#+1, 6] + 16 StirlingS2[n/#, 6], Divisible[#, 3], StirlingS2[n/#+4, 6] - 12 StirlingS2[n/#+3, 6] + 53 StirlingS2[n/#+2, 6] - 102 StirlingS2[n/#+1, 6] + 81 StirlingS2[n/#, 6], Divisible[#, 2], StirlingS2[n/#+3, 6] - 3 StirlingS2[n/#+2, 6] - 8 StirlingS2[n/#+1, 6] + 16 StirlingS2[n/#, 6], True, StirlingS2[n/#, 6]] &], {n, 1, 40}]
mx = 40; Drop[CoefficientList[Series[-Sum[(EulerPhi[d] / d) Which[
  Divisible[d, 60], Log[1 - 6x^d] - Log[1 - 5x^d], Divisible[d, 30],
  (3 Log[1 - 6x^d] - 3 Log[1 - 5x^d] + Log[1 - 2x^d] - Log[1 - x^d]) / 4,
  Divisible[d, 20], (5 Log[1 - 6x^d] - 6 Log[1 - 5x^d] + 2 Log[1 - 3x^d] -
  3 Log[1 - 2x^d]) / 9, Divisible[d, 15], (5 Log[1 - 6x^d] -
  6 Log[1 - 5x^d] + 3 Log[1 - 4x^d] - 4 Log[1 - 3x^d] + 3 Log[1 - 2x^d] -
  6 Log[1 - x^d]) / 16, Divisible[d, 12], (4 Log[1 - 6x^d] -
  4 Log[1 - 5x^d] + Log[1 - x^d]) / 5, Divisible[d, 10], (11 Log[1 - 6x^d] -
  15 Log[1 - 5x^d] + 8 Log[1 - 3x^d] - 3 Log[1 - 2x^d] - 9 Log[1 - x^d]) /
  36, Divisible[d, 6], (11 Log[1 - 6x^d] - 11 Log[1 - 5x^d] +
  5 Log[1 - 2x^d] - Log[1 - x^d]) / 20, Divisible[d, 5], (29 Log[1 - 6x^d] -
  30 Log[1 - 5x^d] + 3 Log[1 - 4x^d] - 4 Log[1 - 3x^d] + 3 Log[1 - 2x^d] -
  30 Log[1 - x^d]) / 144, Divisible[d, 4], (16 Log[1 - 6x^d] -
  21 Log[1 - 5x^d] + 10 Log[1 - 3x^d] - 15 Log[1 - 2x^d] + 9 Log[1 - x^d]) /
  45, Divisible[d, 3], (9 Log[1 - 6x^d] - 14 Log[1 - 5x^d] +
  15 Log[1 - 4x^d] - 20 Log[1 - 3x^d] + 15 Log[1 - 2x^d] -
  14 Log[1 - x^d]) / 80, Divisible[d, 2], (19 Log[1 - 6x^d] -
  39 Log[1 - 5x^d] + 40 Log[1 - 3x^d] - 15 Log[1 - 2x^d] - 9 Log[1 - x^d]) /
  180, True, (Log[1 - 6x^d] - 6 Log[1 - 5x^d] + 15 Log[1 - 4x^d] -
  20 Log[1 - 3x^d] + 15 Log[1 - 2x^d] - 6 Log[1 - x^d]) / 720],
  {d, 1, mx}], {x, 0, mx}], x], 1]
(End)

Crossrefs

Column 6 of A152175.

Cf. A056286, A056293, A056294.

Sequence in context: A127960 A005446 A056307 * A212616 A073980 A034860

Adjacent sequences: A056296 A056297 A056298 * A056300 A056301 A056302

Keyword

nonn

Author

Marks R. Nester

Status

approved