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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

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Author

Marks R. Nester

Keywords

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

Crossrefs

Column 6 of A152175.
Cf. A056286, A056293, A056294.

Programs

  • 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)

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)

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