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!)
A056298 Number of n-bead necklace structures using exactly five different colored beads. 5
0, 0, 0, 0, 1, 3, 20, 136, 773, 4281, 22430, 115100, 577577, 2863227, 14051164, 68515514, 332514803, 1608800691, 7767857090, 37460388596, 180536313547, 869901397479, 4192038616700, 20208367895980 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

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) = A056293(n) - A056292(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+4,5) -

  6*S2(n/d+3,5) + 11*S2(n/d+2,5) - 6*S2(n/d+1,5)) + [d==30 mod 60] *

  (S2(n/d+4,5) - 8*S2(n/d+3,5) + 26*S2(n/d+2,5) - 43*S2(n/d+1,5) +

  30*S2(n/d,5)) + [d==20 mod 60 | d==40 mod 60] * (S2(n/d+4,5) -

  8*S2(n/d+3,5) + 23*S2(n/d+2,5) - 24*S2(n/d+1,5)) + [d==15 mod 60 |

  d==45 mod 60] * (S2(n/d+4,5) - 10*S2(n/d+3,5) + 38*S2(n/d+2,5) -

  65*S2(n/d+1,5) + 45*S2(n/d,5)) + [d mod 60 in {12,24,36,48}] *

  (4*S2(n/d+3,5) - 24*S2(n/d+2,5) + 44*S2(n/d+1,5) - 24*S2(n/d,5)) +

  [d=10 mod 60 | d==50 mod 60] * (S2(n/d+4,5) - 10*S2(n/d+3,5) +

  38*S2(n/d+2,5) - 61*S2(n/d+1,5) + 30*S2(n/d,5)) + [d mod 60 in

  {6,18,42,54}] * (2*S2(n/d+3,5) - 9*S2(n/d+2,5) + 7*S2(n/d+1,5) +

  6*S2(n/d,5)) + [d mod 60 in {5,25,35,55}] * (S2(n/d+4,5) -

  10*S2(n/d+3,5) + 35*S2(n/d+2,5) - 50*S2(n/d+1,5) + 25*S2(n/d,5)) +

  [d mod 60 in {4,8,16,28,32,44,52,56}] * (2*S2(n/d+3,5) - 12*S2(n/d+2,5) +

  26*S2(n/d+1,5) - 24*S2(n/d,5)) + [d mod 60 in {3,9,21,27,33,39,51,57}] *

  (3*S2(n/d+2,5) - 15*S2(n/d+1,5) + 21*S2(n/d,5)) + [d mod 60 in

  {2,14,22,26,34,38,46,58}] * (3*S2(n/d+2,5) - 11*S2(n/d+1,5) +

  6*S2(n/d,5)) + [d mod 60 in {1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,

  59}] * S2(n/d,5)), where S2(n,k) is the Stirling subset number, A008277.

G.f.: -Sum_{d>0} (phi(d) / d) * ([d==0 mod 60] * (log(1-4x^d) -

  log(1-3x^d)) + [d==30 mod 60] * (3*log[1-5x^d) - 3*log(1-4x^d) +

  log(1-x^d)) / 4 + [d==20 mod 60 | d==40 mod 60] * (2*log(1-5x^d) -

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

  [d==15 mod 60 | d==45 mod 60] * (3*log(1-5x^d) - 3*log(1-4x^d) +

  2*log(1-3x^d) - 2*log(1-2x^d) + 3*log(1-x^d)) / 8 + [d mod 60 in

  {12,24,36,48}] * (4*log(1-5x^d) - 5*log(1-4x^d)) / 5 + [d=10 mod 60 |

  d==50 mod 60] * (5*log(1-5x^d) - 5*log(1-4x^d) + 4*log(1-2x^d) -

  log(1-x^d)) / 12 + [d mod 60 in {6,18,42,54}] * (11*log(1-5x^d) -

  15*log(1-4x^d) + 5*log(1-x^d)) / 20 + [d mod 60 in {5,25,35,55}] *

  (5*log(1-5x^d) - log(1-4x^d) + 2*log(1-3x^d) - 2*log(1-2x^d) +

  log(1-x^d)) / 24 + [d mod 60 in {4,8,16,28,32,44,52,56}] *

  (7*log(1-5x^d) - 10*log(1-4x^d) + 5*log(1-2x^d) - 5*log(1-x^d)) /

  15 + [d mod 60 in {3,9,21,27,33,39,51,57}] * (7*log(1-5x^d) -

  15*log(1-4x^d) + 10*log(1-3x^d) - 10*log(1-2x^d) + 15*log(1-x^d)) /

  40 + [d mod 60 in {2,14,22,26,34,38,46,58}] * (13*log(1-5x^d) -

  25*log(1-4x^d) + 20*log(1-2x^d) - 5*log(1-x^d)) / 60 + [d mod 60 in

  {1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,59}] * (log(1-5x^d) -

  5*log(1-4x^d) + 10*log(1-3x^d) - 10*log(1-2x^d) + 5*log(1-x^d)) / 120).

(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, 5],

  {n, 1, 40}] (* after Gilbert and Riordan *)

Table[(1/n) DivisorSum[n, EulerPhi[#] Which[Divisible[#, 60], StirlingS2[n/#+4, 5] - 6 StirlingS2[n/#+3, 5] + 11 StirlingS2[n/#+2, 5] - 6 StirlingS2[n/#+1, 5], Divisible[#, 30], StirlingS2[n/#+4, 5] - 8 StirlingS2[n/#+3, 5] + 26 StirlingS2[n/#+2, 5] - 43 StirlingS2[n/#+1, 5] + 30 StirlingS2[n/#, 5], Divisible[#, 20], StirlingS2[n/#+4, 5] - 8 StirlingS2[n/#+3, 5] + 23 StirlingS2[n/#+2, 5] - 24 StirlingS2[n/#+1, 5], Divisible[#, 15], StirlingS2[n/#+4, 5] - 10 StirlingS2[n/#+3, 5] + 38 StirlingS2[n/#+2, 5] - 65 StirlingS2[n/#+1, 5] + 45 StirlingS2[n/#, 5], Divisible[#, 12], 4 StirlingS2[n/#+3, 5] - 24 StirlingS2[n/#+2, 5] + 44 StirlingS2[n/#+1, 5] - 24 StirlingS2[n/#, 5], Divisible[#, 10], StirlingS2[n/#+4, 5] - 10 StirlingS2[n/#+3, 5] + 38 StirlingS2[n/#+2, 5] - 61 StirlingS2[n/#+1, 5] + 30 StirlingS2[n/#, 5], Divisible[#, 6], 2 StirlingS2[n/#+3, 5] - 9 StirlingS2[n/#+2, 5] + 7 StirlingS2[n/#+1, 5] + 6 StirlingS2[n/#, 5], Divisible[#, 5], StirlingS2[n/#+4, 5] - 10 StirlingS2[n/#+3, 5] + 35 StirlingS2[n/#+2, 5] - 50 StirlingS2[n/#+1, 5] + 25 StirlingS2[n/#, 5], Divisible[#, 4], 2 StirlingS2[n/#+3, 5] - 12 StirlingS2[n/#+2, 5] + 26 StirlingS2[n/#+1, 5] - 24 StirlingS2[n/#, 5], Divisible[#, 3], 3 StirlingS2[n/#+2, 5] - 15 StirlingS2[n/#+1, 5] + 21 StirlingS2[n/#, 5], Divisible[#, 2], 3 StirlingS2[n/#+2, 5] - 11 StirlingS2[n/#+1, 5] + 6 StirlingS2[n/#, 5], True, StirlingS2[n/#, 5]] &], {n, 1, 40}]

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

  Divisible[d, 60], Log[1 - 5x^d] - Log[1 - 4x^d], Divisible[d, 30],

  (3 Log[1 - 5x^d] - 3 Log[1 - 4x^d] + Log[1 - x^d]) / 4, Divisible[d, 20],

  (2 Log[1 - 5x^d] - 2 Log[1 - 4x^d] + Log[1 - 2x^d] - Log[1 - x^d]) / 3,

  Divisible[d, 15], (3 Log[1 - 5x^d] - 3 Log[1 - 4x^d] + 2 Log[1 - 3x^d] -

  2 Log[1 - 2x^d] + 3 Log[1 - x^d]) / 8, Divisible[d, 12],

  (4 Log[1 - 5x^d] - 5 Log[1 - 4x^d]) / 5, Divisible[d, 10],

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

  Divisible[d, 6], (11 Log[1 - 5x^d] - 15 Log[1 - 4x^d] + 5 Log[1 - x^d]) /

  20, Divisible[d, 5], (5 Log[1 - 5x^d] - Log[1 - 4x^d] + 2 Log[1 - 3x^d] -

  2 Log[1 - 2x^d] + Log[1 - x^d]) / 24, Divisible[d, 4], (7 Log[1 - 5x^d] -

  10 Log[1 - 4x^d] + 5 Log[1 - 2x^d] - 5 Log[1 - x^d]) / 15,

  Divisible[d, 3], (7 Log[1 - 5x^d] - 15 Log[1 - 4x^d] + 10 Log[1 - 3x^d] -

  10 Log[1 - 2x^d] + 15 Log[1 - x^d]) / 40, Divisible[d, 2],

  (13 Log[1 - 5x^d] - 25 Log[1 - 4x^d] + 20 Log[1 - 2x^d] -

  5 Log[1 - x^d]) / 60, True, (Log[1 - 5x^d] - 5 Log[1 - 4x^d] +

  10 Log[1 - 3x^d] - 10 Log[1 - 2x^d] + 5 Log[1 - x^d]) / 120], {d, 1, mx}], {x, 0, mx}], x], 1]

(End)

CROSSREFS

Column 5 of A152175.

Cf. A056285, A056292, A056293.

Sequence in context: A000276 A216778 A056306 * A114479 A074574 A267899

Adjacent sequences:  A056295 A056296 A056297 * A056299 A056300 A056301

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 February 21 23:41 EST 2020. Contains 332113 sequences. (Running on oeis4.)