A087854 -id:A087854 - cp's OEIS frontend

A056285 Number of n-bead necklaces with exactly five different colored beads.

0, 0, 0, 0, 24, 300, 2400, 15750, 92680, 510312, 2691600, 13794150, 69309240, 343501500, 1686135376, 8221437000, 39901776360, 193054016840, 932142850800, 4495236798162, 21664357535320, 104388120866100, 503044634004000, 2425003924383900, 11696087875731624

Offset: 1

Marks R. Nester

nonn

Turning over the necklace is not allowed.

			For n=5, the 24 necklaces are A followed by the 24 permutations of BCDE.

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. A000031, A001867, A001868, A001869, A008277.

Column k=5 of A087854.

k=5; Table[k!DivisorSum[n,EulerPhi[#]StirlingS2[n/#,k]&]/n,{n,1,30}] (* Robert A. Russell, Sep 26 2018 *)

a(n) = my(k=5); k!*sumdiv(n, d, eulerphi(d)*stirling(n/d, k, 2))/n; \\ Michel Marcus, Sep 27 2018

a(n) = A001869(n) - 5*A001868(n) + 10*A001867(n) - 10*A000031(n) + 5.
From Robert A. Russell, Sep 26 2018: (Start)
a(n) = (k!/n) 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.: -Sum_{d>0} (phi(d)/d) * Sum_{j} (-1)^(k-j) * C(k,j) * log(1-j x^d), where k=5 is the number of colors. (End)

A056286 Number of n-bead necklaces with exactly six different colored beads.

Original entry on oeis.org

0, 0, 0, 0, 0, 120, 2160, 23940, 211680, 1643544, 11748240, 79419180, 516257280, 3262443120, 20193277104, 123071707080, 741419995680, 4427490147480, 26264144909520, 155018841055596, 911509010154720, 5344538384445120, 31272099902089200, 182707081122261480

Offset: 1

Marks R. Nester

nonn

Turning over the necklace is not allowed.

			For n=6, the 120 necklaces are A followed by the 120 permutations of BCDEF.

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. A000031, A001867, A001868, A001869, A054625.

Column k=6 of A087854.

k=6; Table[k!DivisorSum[n,EulerPhi[#]StirlingS2[n/#,k]&]/n,{n,1,30}] (* Robert A. Russell, Sep 26 2018 *)

a(n) = A054625(n) - 6*A001869(n) + 15*A001868(n) - 20*A001867(n) + 15*A000031(n) - 6.
From Robert A. Russell, Sep 26 2018: (Start)
a(n) = (k!/n) Sum_{d|n} phi(d) S2(n/d,k), where k=6 is the number of colors and S2 is the Stirling subset number A008277.
G.f.: -Sum_{d>0} (phi(d)/d) * Sum_{j} (-1)^(k-j) * C(k,j) * log(1-j x^d), where k=6 is the number of colors. (End)

A330618 Triangle read by rows: T(n,k) is the number of n-bead necklaces using exactly k colors with no adjacent beads having the same color.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 1, 3, 6, 0, 0, 6, 24, 24, 0, 1, 11, 80, 180, 120, 0, 0, 18, 240, 960, 1440, 720, 0, 1, 33, 696, 4410, 11340, 12600, 5040, 0, 0, 58, 1960, 18760, 73920, 137760, 120960, 40320, 0, 1, 105, 5508, 76368, 433944, 1209600, 1753920, 1270080, 362880

Offset: 1

Andrew Howroyd, Dec 20 2019

In the case of n = 1, the single bead is considered to be cyclically adjacent to itself giving T(1,1) = 0. If compatibility with A208535 is wanted then T(1,1) should be 1.

			Triangle begins:
  0;
  0, 1;
  0, 0,  2;
  0, 1,  3,    6;
  0, 0,  6,   24,    24;
  0, 1, 11,   80,   180,   120;
  0, 0, 18,  240,   960,  1440,    720;
  0, 1, 33,  696,  4410, 11340,  12600,   5040;
  0, 0, 58, 1960, 18760, 73920, 137760, 120960, 40320;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Column 3 is A093367.
Row sums are A330620.
Cf. A087854, A208535, A327396, A330341.

\\ here U(n,k) is A208535(n,k) for n > 1.
U(n, k)={sumdiv(n, d, eulerphi(n/d)*(k-1)^d)/n - if(n%2, k-1)}
T(n,k)={sum(j=1, k, (-1)^(k-j)*binomial(k,j)*U(n,j))}

T(n,k) = Sum_{j=1..k} (-1)^(k-j)*binomial(k,j)*A208535(n,j) for n > 1.

T(n,n) = (n-1)! for n > 1.

A238404 Number of ways a prime from A087054 can be decomposed as a sum of the form pq+qr+r*p where p, q and r are distinct primes (p < q < r).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 3, 4, 1, 2, 1, 1, 1, 1, 3, 2, 2, 1, 1, 8, 1, 1, 2, 3, 2, 1, 1, 3, 2, 1, 1, 3, 1, 5, 4, 3, 1, 3, 1, 1, 4, 1, 1, 3, 2, 4, 1, 1, 3, 1, 1, 2, 1, 3, 2, 2, 1, 1, 3, 2, 5, 1, 1, 7, 8, 1, 3, 4, 1, 6, 3, 2, 12, 1, 1, 1, 1, 5, 2, 1, 9, 1, 1, 1, 2, 1, 5, 1, 2, 1, 3, 3, 1, 2, 7, 1

Offset: 1

Jean-François Alcover, Feb 26 2014

nonn

			A087054(5) = 71 = 3*5 + 5*7 + 7*3 = 2*3 + 3*13 + 13*2, therefore a(5) = 2.

Cf. A087053, A087054.

nn = 100; A087854 = Take[Select[ Union[Total[Times @@@ Subsets[#, {2}]] & /@ Subsets[Prime[Range[nn]], {3}]], PrimeQ], nn]; r[n_, p_] := Reduce[p < q < r && p*q+q*r+r*p == n, {q, r}, Primes]; a[n_] := (For[cnt = 0; p = 2, p <= Ceiling[(n-6)/5], p = NextPrime[p], rnp = r[n, p]; If[rnp =!= False, Which[rnp[[0]] === And, Print["n = ", n, " ", {p, q, r} /. ToRules[rnp]]; cnt++, rnp[[0]] === Or, Print["n = ", n, " ", {p, q, r} /. {ToRules[rnp]}]; cnt += Length[rnp], True, Print["error: n = ", n, " ", rnp]]]]; cnt); Reap[Do[ap = a[p]; If[ap > 0, Sow[ap]], {p, A087854}]][[2, 1]] (* after Harvey P. Dale *)

A337827 a(n) is the number of 2n-bead necklaces with exactly n different colored beads.

Original entry on oeis.org

1, 4, 91, 5106, 510312, 79419180, 17758541160, 5397245416080, 2140495978440960, 1073686615987184640, 664582969579048732800, 497566995304189676342400, 443212653988584642449548800, 463237380681508395323231270400, 561422444732790213860755013145600, 780983354978825959061219179885824000

Offset: 1

Yves-Loic Martin, Sep 24 2020

nonn

			a(2) = 4, corresponding to the necklaces WBBB, WBWB, WWBB, and WWWB.

Wikipedia, Necklace combinatorics

Cf. A087854, A007820.

Table[n! * (StirlingS2[2*n, n] + 1) / (2*n), {n, 1, 16}] (* Amiram Eldar, Sep 25 2020 *)

T(n,k) = (k!/n) * sumdiv(n,d, eulerphi(d) * stirling(n/d, k,2)); \\ A087854
vector(22,n,T(2*n,n)) \\ Joerg Arndt, Sep 25 2020

a(n) = A087854(2*n,n) = (n!/(2*n)) * Sum_{d|2*n} phi(d) * S2(2*n/d, n) where S2(n,k) are the Stirling numbers of the second kind.

a(n) = (n!/(2*n))*(S2(2*n, n)+1) since S2(n, n) = 1 and S2(2*n/d, n) = 0 if d>2.

Terms a(6) and beyond from Joerg Arndt, Sep 25 2020

cp's OEIS Frontend

A056285 Number of n-bead necklaces with exactly five different colored beads.

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A056286 Number of n-bead necklaces with exactly six different colored beads.

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A330618 Triangle read by rows: T(n,k) is the number of n-bead necklaces using exactly k colors with no adjacent beads having the same color.

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A238404 Number of ways a prime from A087054 can be decomposed as a sum of the form p*q+q*r+r*p where p, q and r are distinct primes (p < q < r).

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A337827 a(n) is the number of 2n-bead necklaces with exactly n different colored beads.

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Extensions

A238404 Number of ways a prime from A087054 can be decomposed as a sum of the form pq+qr+r*p where p, q and r are distinct primes (p < q < r).