A056285 -id:A056285 - cp's OEIS frontend

A087854 Triangle read by rows: T(n,k) is the number of n-bead necklaces with exactly k different colored beads.

1, 1, 1, 1, 2, 2, 1, 4, 9, 6, 1, 6, 30, 48, 24, 1, 12, 91, 260, 300, 120, 1, 18, 258, 1200, 2400, 2160, 720, 1, 34, 729, 5106, 15750, 23940, 17640, 5040, 1, 58, 2018, 20720, 92680, 211680, 258720, 161280, 40320, 1, 106, 5613, 81876, 510312, 1643544, 2963520, 3024000, 1632960, 362880

Offset: 1

Philippe Deléham

Equivalently, T(n,k) is the number of sequences (words) of length n on an alphabet of k letters where each letter of the alphabet occurs at least once in the sequence. Two sequences are considered equivalent if one can be obtained from the other by a cyclic shift of the letters. Cf. A054631 where the surjective restriction is removed. - Geoffrey Critzer, Jun 18 2013

Robert A. Russell's g.f. for column k >= 1 (in the Formula section below) can be proved by integrating both sides of the formula Sum_{n>=1} S2(n, k)*x^(n-1) = x^(k-1)/((1 - x)* (1 - 2*x) * (1 - 3*x) * ... * (1 - k*x)) w.r.t. x. A variation of this identity (valid for |x| < 1/k) can be found in the Formula section of A008277. - Petros Hadjicostas, Aug 20 2019

			The triangle begins with T(1,1):
  1;
  1,   1;
  1,   2,    2;
  1,   4,    9,     6;
  1,   6,   30,    48,     24;
  1,  12,   91,   260,    300,     120;
  1,  18,  258,  1200,   2400,    2160,     720;
  1,  34,  729,  5106,  15750,   23940,   17640,    5040;
  1,  58, 2018, 20720,  92680,  211680,  258720,  161280,   40320;
  1, 106, 5613, 81876, 510312, 1643544, 2963520, 3024000, 1632960, 362880;
  ...
For T(4,2) = 4, the necklaces are AAAB, AABB, ABAB, and ABBB.
For T(4,4) = 6, the necklaces are ABCD, ABDC, ACBD, ACDB, ADBC, and ADCB.

Alois P. Heinz, Rows n = 1..141, flattened

Columns 1-6: A057427, A052823, A056283, A056284, A056285, A056286.
Diagonals: A000142 and A074143.
Row sums: A019536.
Cf. A000010 (Euler totient phi function), A008277 (Stirling2 numbers), A075195 (table of Jablonski).
Cf. A254040, A273891.

with(numtheory):
T:= (n, k)-> (k!/n) *add(phi(d) *Stirling2(n/d, k), d=divisors(n)):
seq(seq(T(n,k), k=1..n), n=1..12);  # Alois P. Heinz, Jun 19 2013

Table[Table[Sum[EulerPhi[d]*StirlingS2[n/d,k]k!,{d,Divisors[n]}]/n,{k,1,n}],{n,1,10}]//Grid (* Geoffrey Critzer, Jun 18 2013 *)

T(n, k) = (k!/n) * sumdiv(n, d, eulerphi(d) * stirling(n/d, k, 2)); \\ Joerg Arndt, Sep 25 2020

T(n,k) = Sum_{i=0..k-1} (-1)^i * C(k,i) * A075195(n,k-i); A075195 = Jablonski's table.
T(n,k) = (k!/n) * Sum_{d|n} phi(d) * S2(n/d, k), where S2(n,k) = Stirling numbers of 2nd kind A008277.
G.f. for column k: -Sum_{d>0} (phi(d)/d) * Sum_{j = 1..k} (-1)^(k-j) * C(k,j) * log(1 - j * x^d). - Robert A. Russell, Sep 26 2018
T(n,k) = Sum_{d|n} A254040(d, k) for n, k >= 1. - Petros Hadjicostas, Aug 19 2019

Formula section edited by Petros Hadjicostas, Aug 20 2019

A294687 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly five colors under translational symmetry, 1 <= k <= n.

Original entry on oeis.org

0, 0, 0, 0, 300, 92680, 0, 15750, 13794150, 8221452750, 24, 510312, 1686135376, 4495236798162, 11696087875731720, 300, 13794450, 193054017440, 2425003938178050, 30852000867277668428, 403564024914127655401650, 2400, 343501500, 21664357535320, 1317601563731383350, 82985159653854019928352, 5411356249329837891442095560

Offset: 1

Marko Riedel, Nov 06 2017

Colors are not being permuted, i.e., Power Group Enumeration does not apply here.

			Triangle begins:
   0;
   0,      0;
   0,    300,      92680;
   0,  15750,   13794150,    8221452750;
  24, 510312, 1686135376, 4495236798162, 11696087875731720;
  ...

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.

Andrew Howroyd, Table of n, a(n) for n = 1..325 (first 25 rows).
Marko Riedel et al., Burnside lemma and translational symmetries of the torus.

Main diagonal is A376825.

Cf. A294684, A294685, A294686, A294791, A294792, A294793, A294794. T(n,1) is A056285.

T(n,m)=my(k=5); k!*sumdiv(n, d, sumdiv(m, e, eulerphi(d) * eulerphi(e) * stirling(n*m/lcm(d,e), k, 2) ))/(n*m) \\ Andrew Howroyd, Oct 05 2024

T(n,k) = (Q!/(n*k))*(Sum_{d|n} Sum_{f|k} phi(d) phi(f) S(gcd(d,f)*(n/d)*(k/f), Q)) with Q=5 and S(n,k) Stirling numbers of the second kind.

A056298 Number of n-bead necklace structures using exactly five different colored beads.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 20, 136, 773, 4281, 22430, 115100, 577577, 2863227, 14051164, 68515514, 332514803, 1608800691, 7767857090, 37460388596, 180536313547, 869901397479, 4192038616700, 20208367895980

Offset: 1

Marks R. Nester

nonn

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

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]

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

Column 5 of A152175.

Cf. A056285, A056292, A056293.

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)

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)

A056491 Number of periodic palindromes using exactly five different symbols.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 60, 120, 960, 1800, 9300, 16800, 71400, 126000, 480060, 834120, 2968560, 5103000, 17355300, 29607600, 97567800, 165528000, 533274060, 901020120, 2855012160, 4809004200, 15050517300, 25292030400, 78417448200, 131542866000, 404936532060

Offset: 1

Marks R. Nester

			For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.
There are 120 permutations of the five letters used in ABACDEDC.  These 120 arrangements can be paired up with a half turn (e.g., ABACDEDC-DEDCABAC) to arrive at the 60 arrangements for n=8. - _Robert A. Russell_, Sep 26 2018

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]

Muniru A Asiru, Table of n, a(n) for n = 1..600
Index entries for linear recurrences with constant coefficients, signature (1,14,-14,-71,71,154,-154,-120, 120).

Cf. A056456.

Column 5 of A305540.

a:=[0,0,0,0,0,0,0,60,120];; for n in [10..35] do a[n]:=a[n-1]+14*a[n-2]-14*a[n-3]-71*a[n-4]+71*a[n-5]+154*a[n-6]-154*a[n-7]-120*a[n-8]+120*a[n-9]; od; a; # Muniru A Asiru, Sep 26 2018

m:=50; R:=PowerSeriesRing(Integers(), m); [0, 0, 0, 0, 0, 0, 0] cat Coefficients(R!(-60*x^8*(x+1)/((x-1)*(2*x-1)*(2*x+1)*(2*x^2-1)*(3*x^2-1)*(5*x^2-1)))); // G. C. Greubel, Oct 13 2018

with(combinat):  a:=n->(factorial(5)/2)*(Stirling2(floor((n+1)/2),5)+Stirling2(ceil((n+1)/2),5)): seq(a(n),n=1..35); # Muniru A Asiru, Sep 26 2018

k = 5; Table[(k!/2) (StirlingS2[Floor[(n + 1)/2], k] +
StirlingS2[Ceiling[(n + 1)/2], k]), {n, 1, 40}] (* Robert A. Russell, Jun 05 2018 *)
LinearRecurrence[{1, 14, -14, -71, 71, 154, -154, -120, 120}, {0, 0,
0, 0, 0, 0, 0, 60, 120}, 40] (* Robert A. Russell, Sep 29 2018 *)

a(n) = my(k=5); (k!/2)*(stirling(floor((n+1)/2), k, 2) + stirling(ceil((n+1)/2), k, 2)); \\ Michel Marcus, Jun 05 2018

a(n) = 2*A056345(n) - A056285(n).
G.f.: -60*x^8*(x+1)/((x-1)*(2*x-1)*(2*x+1)*(2*x^2-1)*(3*x^2-1)*(5*x^2-1)). - Colin Barker, Jul 08 2012
a(n) = (k!/2)*(S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)), with k=5 different colors used and where S2(n,k) is the Stirling subset number A008277. - Robert A. Russell, Jun 05 2018
a(n) = a(n-1) + 14*a(n-2) - 14*a(n-3) - 71*a(n-4) + 71*a(n-5) + 154*a(n-6) - 154*a(n-7) - 120*a(n-8) + 120*a(n-9). - Muniru A Asiru, Sep 26 2018

A056290 Number of primitive (period n) n-bead necklaces with exactly five different colored beads.

Original entry on oeis.org

0, 0, 0, 0, 24, 300, 2400, 15750, 92680, 510288, 2691600, 13793850, 69309240, 343499100, 1686135352, 8221421250, 39901776360, 193053923860, 932142850800, 4495236287850, 21664357532920, 104388118174500, 503044634004000, 2425003910574000, 11696087875731600

Offset: 1

Marks R. Nester

nonn

Turning over the necklace is not allowed.

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]

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A001692.

Column k=5 of A254040.

with(numtheory):
b:= proc(n, k) option remember; `if`(n=0, 1,
      add(mobius(n/d)*k^d, d=divisors(n))/n)
    end:
a:= n-> add(b(n, 5-j)*binomial(5, j)*(-1)^j, j=0..5):
seq(a(n), n=1..30);  # Alois P. Heinz, Jan 25 2015

b[n_, k_] := b[n, k] = If[n==0, 1, DivisorSum[n, MoebiusMu[n/#]*k^# &]/n];
a[n_] := Sum[b[n, 5 - j]*Binomial[5, j]*(-1)^j, {j, 0, 5}];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jun 06 2018, after Alois P. Heinz *)

sum mu(d)*A056285(n/d) where d|n.

A056345 Number of bracelets of length n using exactly five different colored beads.

Original entry on oeis.org

0, 0, 0, 0, 12, 150, 1200, 7905, 46400, 255636, 1346700, 6901725, 34663020, 171786450, 843130688, 4110958530, 19951305240, 96528492700, 466073976900, 2247627076731, 10832193571460, 52194109216950

Offset: 1

Marks R. Nester

nonn

Turning over will not create a new bracelet.

			For a(5)=12, pair up the 24 permutations of BCDE, each with its reverse, such as BCDE-EDCB.  Precede the first of each pair with an A, such as ABCDE.  These are the 12 arrangements, all chiral.  If we precede the second of each pair with an A, such as AEDCB, we get the chiral partner of each. - _Robert A. Russell_, Sep 27 2018

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]

Column 5 of A273891.

Equals (A056285 + A056491) / 2 = A056285 - A305544 = A305544 + A056491.

t[n_, k_] := (For[t1 = 0; d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*k^(n/d)]]; If[EvenQ[n], (t1 + (n/2)*(1 + k)*k^(n/2))/(2*n), (t1 + n*k^((n + 1)/2))/(2*n)]);
T[n_, k_] := Sum[(-1)^i*Binomial[k, i]*t[n, k - i], {i, 0, k - 1}];
a[n_] := T[n, 5];
Array[a, 22] (* Jean-François Alcover, Nov 05 2017, after Andrew Howroyd *)
k=5; Table[k! DivisorSum[n, EulerPhi[#] StirlingS2[n/#,k]&]/(2n) + k!(StirlingS2[Floor[(n+1)/2], k] + StirlingS2[Ceiling[(n+1)/2], k])/4, {n,1,30}] (* Robert A. Russell, Sep 27 2018 *)

a(n) = A032276(n) - 5*A032275(n) + 10*A027671(n) - 10*A000029(n) + 5.
From Robert A. Russell, Sep 27 2018: (Start)
a(n) = (k!/4) * (S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)) + (k!/2n) * Sum_{d|n} phi(d) * S2(n/d,k), where k=5 is the number of colors and S2 is the Stirling subset number A008277.
G.f.: (k!/4) * x^(2k-2) * (1+x)^2 / Product_{i=1..k} (1-i x^2) - Sum_{d>0} (phi(d)/2d) * Sum_{j} (-1)^(k-j) * C(k,j) * log(1-j x^d), where k=5 is the number of colors. (End)

cp's OEIS Frontend

A087854 Triangle read by rows: T(n,k) is the number of n-bead necklaces with exactly k different colored beads.

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A294687 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly five colors under translational symmetry, 1 <= k <= n.

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A056298 Number of n-bead necklace structures using exactly five different colored beads.

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A056491 Number of periodic palindromes using exactly five different symbols.

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A056290 Number of primitive (period n) n-bead necklaces with exactly five different colored beads.

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A056345 Number of bracelets of length n using exactly five different colored beads.

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