A330341 -id:A330341 - cp's OEIS frontend

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.

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.

A330621 Number of length n bracelets with entries covering an initial interval of positive integers and no adjacent entries equal.

Original entry on oeis.org

0, 1, 1, 7, 27, 207, 1689, 17137, 196869, 2556856, 36878013, 585247590, 10131891315, 190024056601, 3838053182983, 83057105368627, 1917217162193175, 47021314781221603, 1221073517359584357, 33471097453271690668, 965771726172667547339, 29259595679585441629303

Offset: 1

Andrew Howroyd, Dec 20 2019

nonn

			Case n=4: there are the following 7 bracelets:
  1212,
  1213, 1232, 1323,
  1234, 1243, 1324.

Andrew Howroyd, Table of n, a(n) for n = 1..200

Row sums of A330341.

Cf. A208544.

\\ here U(n, k) is A208544(n, k) for n > 1.
U(n, k) = (sumdiv(n, d, eulerphi(n/d)*(k-1)^d)/n + if(n%2, 1-k, k*(k-1)^(n/2)/2))/2;
a(n)={if(n<1, n==0, sum(j=1, n, U(n,j)*sum(k=j, n, (-1)^(k-j)*binomial(k, j))))}

A330632 Number of n-bead bracelets using exactly three colors with no adjacent beads having the same color.

Original entry on oeis.org

0, 0, 1, 3, 3, 10, 9, 27, 29, 75, 93, 221, 315, 684, 1095, 2247, 3855, 7682, 13797, 27009, 49939, 96906, 182361, 352695, 671091, 1296855, 2485533, 4806075, 9256395, 17920857, 34636833, 67159047, 130150587, 252745365, 490853415, 954637555, 1857283155

Offset: 1

Andrew Howroyd, Dec 21 2019

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..500

Column 3 of A330341.

Cf. A208539.

cp's OEIS Frontend

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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A330621 Number of length n bracelets with entries covering an initial interval of positive integers and no adjacent entries equal.

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Links

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Programs

PARI

A330632 Number of n-bead bracelets using exactly three colors with no adjacent beads having the same color.

Views

Author

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Crossrefs