A129526 -id:A129526 - cp's OEIS frontend

A250114 Number of orbits of size n in vertex graph of Lucas cube Lambda_n.

1, 1, 1, 1, 2, 2, 4, 5, 6, 9, 12, 15, 20, 28, 30, 47, 54, 79, 88, 131, 138, 222, 232, 356, 374, 598, 602, 967, 986, 1581, 1596, 2573, 2570, 4203, 4174, 6779, 6764, 11058, 10924, 17872, 17710, 28995, 28656, 46920, 46328, 76068, 75024, 122946, 121388, 199199

Offset: 1

N. J. A. Sloane, Nov 19 2014

nonn

Lars Blomberg, Table of n, a(n) for n = 1..1000
A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar and M. Petkovsek, Vertex and edge orbits of Fibonacci and Lucas cubes, 2014; See Table 4.
A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar, et al., Orbits of Fibonacci and Lucas cubes, dihedral transformations, and asymmetric strings, 2014.

Cf. A129526, A250115.

More terms from Lars Blomberg, Dec 05 2016

A250115 Number of orbits of size 2n in vertex graph of Lucas cube Lambda_n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 10, 15, 30, 44, 78, 119, 202, 310, 513, 786, 1277, 1977, 3168, 4916, 7831, 12199, 19332, 30208, 47756, 74808, 118124, 185415, 292673, 460270, 726598, 1144499, 1807638, 2851356, 4506370, 7117298, 11256870, 17799183, 28173716

Offset: 1

N. J. A. Sloane, Nov 19 2014

nonn

Lars Blomberg, Table of n, a(n) for n = 1..1000
A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar and M. Petkovsek, Vertex and edge orbits of Fibonacci and Lucas cubes, 2014; See Table 4.
A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar, et al., Orbits of Fibonacci and Lucas cubes, dihedral transformations, and asymmetric strings, 2014.

Cf. A129526, A250114.

More terms from Lars Blomberg, Dec 05 2016

A143654 Array T(n,k) read by rows: number of binary bracelets with n beads, k of them 0, with 00 prohibited, (n >= 2, 0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 4, 3, 1, 1, 1, 4, 5, 4, 1, 1, 1, 5, 7, 8, 3, 1, 1, 1, 5, 8, 10, 5, 1, 1, 1, 6, 10, 16, 10, 4, 1, 1, 1, 6, 12, 20, 16, 7, 1, 1, 1, 7, 14, 29, 26, 16, 4, 1, 1, 1, 7, 16, 35, 38, 26, 8, 1, 1, 1, 8, 19, 47, 57, 50

Offset: 2

Washington Bomfim, Aug 28 2008

The initial columns give A057427, A057427, A004526, A069905, A005232, A032279, A005513, A032280, A005514, A032281, A005515, A032282, A005516. Row sums give A129526.
A binary bracelet with n beads, k of them 0, with 00 prohibited has from 0 to floor(n/2) beads 0, i.e., 0 <= k <= floor(n/2). If n is even, the bracelet 0101...01 with n/2 beads of each kind does not have 00 and we cannot change any 1 of it to a 0. If n is odd we cannot change a 1 to a 0 in the bracelet 0101...011 with (n-1)/2 beads 0.
The number of binary bracelets with n beads, 0 <= k <= floor(n/2) of them 0 with 00 prohibited, is equal to the number of binary bracelets with n-k beads, k of them 0. See below.
Let B be a binary bracelet with n-k beads, k of them 0. If we insert one 1 (circularly) after a 0 of B, we obtain a bracelet with n-k+1 beads, k of them 0.
If we do this insertion k times, each time after a distinct 0 of B, we obtain a bracelet with n = n-k+k beads, k of them 0, with 00 prohibited.
On the contrary, Let B be a binary bracelet with n beads, k of them 0, with 00 prohibited. If we remove from B one 1 that is after a 0, we obtain a bracelet of n-1 beads, k of them 0. (If not and we undo the removal, the configuration obtained cannot be a bracelet and this is absurd.) If we repeat this removal k times, after each distinct bead 0, we obtain a bracelet with n-k beads, k of them 0.

			Array begins
1 1
1 1
1 1 1
1 1 1
1 1 2 1
1 1 2 1
1 1 3 2 1
1 1 3 3 1
1 1 4 4 3 1
...
A129526(10) = A057427(10) + A057427(9) + A004526(8) + A069905(7) + A005232(6) +
A032279(5) = 1+1+4+4+3+1 = 14.

Cf. A057427, A004526, A069905, A005232, A032279, A005513, A032280, A005514, A032281, A005515, A032282, A005516. Row sums of array give A129526.

A246998 Multiplicity of the zero at x=1 of the characteristic polynomial P_n^{C0+}.

Original entry on oeis.org

1, 2, 2, 3, 3, 5, 5, 8, 9, 14, 16, 26, 31, 49, 64, 99, 133, 209, 291, 455, 657, 1022, 1510, 2359, 3545, 5536, 8442, 13201, 20319, 31836, 49353, 77436, 120711, 189674, 296854, 467160, 733363, 1155647, 1818594, 2869378, 4524081, 7146483

Offset: 1

N. J. A. Sloane, Sep 18 2014

nonn

Appears to duplicate A129526. - R. J. Mathar, Sep 22 2014

A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar, et al., Orbits of Fibonacci and Lucas cubes, dihedral transformations, and asymmetric strings, 2014.
M. Assis, J. L. Jacobsen, I. Jensen, J.-M. Maillard and B. M. McCoy, Integrability vs non-integrability: Hard hexagons and hard squares compared, arXiv preprint 1406.5566 [math-ph], 2014. See Eq. A.15.

The initial terms look wrong to me. I have written to the authors.

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A250114 Number of orbits of size n in vertex graph of Lucas cube Lambda_n.

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A250115 Number of orbits of size 2n in vertex graph of Lucas cube Lambda_n.

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A143654 Array T(n,k) read by rows: number of binary bracelets with n beads, k of them 0, with 00 prohibited, (n >= 2, 0 <= k <= floor(n/2)).

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Crossrefs

A246998 Multiplicity of the zero at x=1 of the characteristic polynomial P_n^{C0+}.

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