A351359 -id:A351359 - cp's OEIS frontend

A351357 Irregular triangle read by rows T(n,k): number of binary necklaces with n beads, k of them black, and at least three consecutive black beads.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 2, 1, 1, 0, 0, 0, 1, 3, 3, 1, 1, 0, 0, 0, 1, 4, 6, 4, 1, 1, 0, 0, 0, 1, 5, 10, 9, 4, 1, 1, 0, 0, 0, 1, 6, 15, 19, 12, 5, 1, 1, 0, 0, 0, 1, 7, 21, 33, 29, 15, 5, 1, 1, 0, 0, 0, 1, 8, 28, 54, 60, 42, 19, 6, 1, 1, 0, 0, 0, 1, 9, 36, 81, 111, 96, 55, 22, 6, 1, 1

Offset: 1

Marko Riedel, Feb 08 2022

			Triangle begins:
  0, 0;
  0, 0, 0;
  0, 0, 0, 1;
  0, 0, 0, 1, 1;
  0, 0, 0, 1, 1, 1;
  0, 0, 0, 1, 2, 1, 1;
  0, 0, 0, 1, 3, 3, 1, 1;
  0, 0, 0, 1, 4, 6, 4, 1, 1;

Marko R. Riedel et al., Necklaces made of black and white beads, Math StackExchange, Feb 2022.

Row sums are A351360.

Cf. A351356, A351358, A351359, A351361.

A351358 Irregular triangle read by rows T(n,k): number of binary necklaces with n beads, k of them black, and at least four consecutive black beads.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 2, 1, 1, 0, 0, 0, 0, 1, 3, 3, 1, 1, 0, 0, 0, 0, 1, 4, 6, 4, 1, 1, 0, 0, 0, 0, 1, 5, 10, 10, 5, 1, 1, 0, 0, 0, 0, 1, 6, 15, 20, 14, 5, 1, 1, 0, 0, 0, 0, 1, 7, 21, 35, 34, 18, 6, 1, 1, 0, 0, 0, 0, 1, 8, 28, 56, 68, 50, 22, 6, 1, 1

Offset: 1

Marko Riedel, Feb 08 2022

			Triangle begins:
  0, 0;
  0, 0, 0;
  0, 0, 0, 0;
  0, 0, 0, 0, 1;
  0, 0, 0, 0, 1, 1;
  0, 0, 0, 0, 1, 1, 1;
  0, 0, 0, 0, 1, 2, 1, 1;
  0, 0, 0, 0, 1, 3, 3, 1, 1;

Marko R. Riedel et al., Necklaces made of black and white beads, Math StackExchange, Feb 2022.

Cf. A351356, A351357, A351359, A351360, A351361.

Row sums are A351361.

A351360 Number of binary necklaces with n beads and at least three consecutive black beads.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 9, 17, 31, 60, 113, 220, 419, 813, 1565, 3033, 5855, 11358, 21977, 42644, 82675, 160517, 311609, 605551, 1176871, 2288964, 4453237, 8669002, 16881859, 32892459, 64112225, 125022545, 243898335, 476011102, 929387453, 1815322932

Offset: 1

Marko Riedel, Feb 08 2022

nonn

Marko R. Riedel et al., Necklaces made of black and white beads, Math StackExchange, Feb 2022.

Cf. A351356, A351357, A351358, A351359, A351361.

a(n) = A000031(n) - A093305(n).

A351361 Number of binary necklaces with n beads and at least four consecutive black beads.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 5, 9, 17, 33, 63, 124, 241, 475, 930, 1831, 3593, 7077, 13905, 27377, 53847, 106006, 208601, 410723, 808514, 1592123, 3135085, 6174695, 12161761, 23957945, 47198847, 92997768, 183253045, 361145269, 711793554, 1403055509, 2765913617

Offset: 1

Marko Riedel, Feb 08 2022

nonn

Marko R. Riedel et al., Necklaces made of black and white beads, Math StackExchange, Feb 2022.

Cf. A351356, A351357, A351358, A351359, A351360.

a(n) = A000031(n) - A280218(n).

A351356 Triangle read by rows T(n,k): number of binary necklaces with n beads, k of them black, and at least two consecutive black beads.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 1, 1, 0, 0, 1, 3, 3, 1, 1, 0, 0, 1, 4, 5, 3, 1, 1, 0, 0, 1, 5, 9, 7, 4, 1, 1, 0, 0, 1, 6, 13, 14, 10, 4, 1, 1, 0, 0, 1, 7, 19, 25, 22, 12, 5, 1, 1, 0, 0, 1, 8, 25, 41, 42, 30, 15, 5, 1, 1, 0, 0, 1, 9, 33, 63, 79, 66, 43, 19, 6, 1, 1, 0, 0, 1, 10, 41, 92, 131, 132, 99, 55, 22, 6, 1, 1

Offset: 1

Marko Riedel, Feb 08 2022

			Triangle begins:
  0;
  0, 0;
  0, 1, 0;
  0, 1, 1, 0;
  0, 1, 1, 1, 0;
  0, 1, 2, 1, 1, 0;
  0, 1, 3, 3, 1, 1, 0;
  0, 1, 4, 5, 3, 1, 1, 0;
  0, 1, 5, 9, 7, 4, 1, 1, 0;
  ...

Marko R. Riedel et al., Necklaces made of black and white beads, Math StackExchange, Feb 2022.
Marko R. Riedel, Maple code for sequence from cited link.

Row sums are A351359.

Cf. A351357, A351358, A351360, A351361.

A351363 Number of necklaces with n black and n white beads with at least four consecutive black beads.

Original entry on oeis.org

0, 0, 0, 1, 5, 21, 84, 327, 1251, 4760, 18018, 68104, 257088, 970480, 3664070, 13842215, 52331695, 198021936, 750027090, 2843686271, 10792866315, 41006110365, 155961264880, 593797895358, 2263126224780, 8634161689920, 32973466762098, 126047166461574, 482298648250416, 1847156083546316

Offset: 1

Marko Riedel, Feb 08 2022

nonn

Marko R. Riedel et al., Necklaces made of black and white beads, Math StackExchange, Feb 2022.

Cf. A351359, A351357, A351358, A351359, A351360, A351361.

a(n) = A003239(n) - A351364(n).

A351364 Number of necklaces with n black and n white beads that avoid the pattern BBBB.

Original entry on oeis.org

1, 2, 4, 9, 21, 59, 162, 483, 1453, 4492, 14048, 44616, 142936, 462380, 1506534, 4941955, 16303783, 54066560, 180111432, 602481589, 2022797529, 6814336663, 23026359634, 78027238290, 265085903996, 902733374480, 3080967048004, 10536594982790, 36102498293396, 123920278459564

Offset: 1

Marko Riedel, Feb 08 2022

nonn

Marko R. Riedel et al., Necklaces made of black and white beads, Math StackExchange, Feb 2022.

Cf. A351359, A351357, A351358, A351359, A351360, A351361.

a(n) = A003239(n) - A351363(n).

cp's OEIS Frontend

A351357 Irregular triangle read by rows T(n,k): number of binary necklaces with n beads, k of them black, and at least three consecutive black beads.

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A351358 Irregular triangle read by rows T(n,k): number of binary necklaces with n beads, k of them black, and at least four consecutive black beads.

Views

Author

Keywords

Examples

Links

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Formula

A351360 Number of binary necklaces with n beads and at least three consecutive black beads.

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A351361 Number of binary necklaces with n beads and at least four consecutive black beads.

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A351356 Triangle read by rows T(n,k): number of binary necklaces with n beads, k of them black, and at least two consecutive black beads.

Views

Author

Keywords

Examples

Links

Crossrefs

A351363 Number of necklaces with n black and n white beads with at least four consecutive black beads.

Views

Author

Keywords

Links

Crossrefs

Formula

A351364 Number of necklaces with n black and n white beads that avoid the pattern BBBB.

Views

Author

Keywords

Links

Crossrefs

Formula