A209032 -id:A209032 - cp's OEIS frontend

A209025 Number of n-bead necklaces labeled with numbers -1..1 allowing reversal, with sum zero and first differences in -1..1.

1, 1, 1, 2, 2, 4, 6, 13, 21, 45, 83, 181, 362, 794, 1676, 3703, 8049, 17925, 39679, 88980, 199308, 449714, 1015574, 2303940, 5234731, 11931095, 27239911, 62337821, 142894336, 328168558, 754815704, 1738888333, 4011453913, 9266702091, 21433240903, 49633242760

Offset: 1

R. H. Hardin, Mar 04 2012

nonn

			All solutions for n=6:
..0...-1...-1...-1
..0....0...-1....0
..0....0....0....0
..0....1....1....0
..0....0....1....1
..0....0....0....0

Andrew Howroyd, Table of n, a(n) for n = 1..100

Column 1 of A209032.

a(33)-a(36) from Andrew Howroyd, Mar 19 2017

A209026 Number of n-bead necklaces labeled with numbers -2..2 allowing reversal, with sum zero and first differences in -2..2.

Original entry on oeis.org

1, 2, 2, 6, 11, 33, 86, 278, 873, 2938, 9904, 34321, 119673, 422725, 1502590, 5381671, 19377063, 70138385, 254978018, 930709334, 3409356533, 12530190100, 46188325542, 170723964752, 632624262841, 2349655631568, 8745690221340, 32617408075856

Offset: 1

R. H. Hardin, Mar 04 2012

nonn

			Some solutions for n=6:
..0...-1...-1...-2...-2...-2...-1...-1...-2...-1...-1...-1...-2...-1...-1...-1
..0....0....0....0....0...-1...-1...-1...-1...-1...-1....0...-1...-1....0....0
..0....0....0....1....0....1....0....0....1...-1....1....1....1....0...-1....1
..0....1....0....0....1....0....0....0....2....1....0...-1....1....1....0...-1
..0...-1....0....1....1....2....1....2....1....1....0....0....1....0....2....1
..0....1....1....0....0....0....1....0...-1....1....1....1....0....1....0....0

Andrew Howroyd, Table of n, a(n) for n = 1..100

Column 2 of A209032.

a(22)-a(28) from Andrew Howroyd, Mar 19 2017

A209027 Number of n-bead necklaces labeled with numbers -3..3 allowing reversal, with sum zero and first differences in -3..3.

Original entry on oeis.org

1, 2, 4, 12, 34, 144, 576, 2613, 11841, 55773, 265095, 1280476, 6238246, 30674021, 151874427, 756842052, 3792315084, 19096602857, 96586072494, 490453174481, 2499410534082, 12778951549191, 65530990963959, 336965088080673, 1737060054201011, 8975377470866966

Offset: 1

R. H. Hardin, Mar 04 2012

nonn

			Some solutions for n=6:
.-2...-3...-3...-3...-2...-1...-1...-3...-3...-3...-2...-1...-3...-1...-1...-1
..0...-2...-1...-2...-1...-1....1....0....0...-1....0....0...-2...-1...-1...-1
..0....1....1....1....0....2...-1...-1....0....0....2...-1....1...-1....0....0
..2....2....2....1....0...-1....1....2....0....2....0....0....3....2....1...-1
..0....2....1....3....3...-1...-1....2....3....2...-1....0....2...-1....1....2
..0....0....0....0....0....2....1....0....0....0....1....2...-1....2....0....1

Andrew Howroyd, Table of n, a(n) for n = 1..100

Column 3 of A209032.

a(19)-a(26) from Andrew Howroyd, Mar 19 2017

A209028 Number of n-bead necklaces labeled with numbers -4..4 allowing reversal, with sum zero and first differences in -4..4.

Original entry on oeis.org

1, 3, 5, 23, 88, 471, 2517, 14611, 86014, 519574, 3177405, 19681214, 123050881, 775868013, 4926859869, 31483531830, 202298489710, 1306289630067, 8472236548514, 55167003603285, 360512087413974, 2363623888860990, 15542899182846478, 102488033365084376

Offset: 1

R. H. Hardin, Mar 04 2012

nonn

			Some solutions for n=6:
.-1...-1...-3...-2...-1...-2...-4...-1...-1...-3...-3...-2...-2...-2...-3...-2
..0...-1...-3....0...-1....1....0...-1....0...-1...-1....0...-1....1...-1....0
.-1...-1...-1...-1...-1....1....2....0...-1....0....2....0...-1...-1....2....3
..1....0....2....2....0...-2....0....0....1....3...-1....0....1....2....0...-1
..0....1....4...-1....3....1....2....0...-1....0....3....2....2...-1....1...-1
..1....2....1....2....0....1....0....2....2....1....0....0....1....1....1....1

Andrew Howroyd, Table of n, a(n) for n = 1..100

Column 4 of A209032.

a(17)-a(24) from Andrew Howroyd, Mar 19 2017

A209029 Number of n-bead necklaces labeled with numbers -5..5 allowing reversal, with sum zero and first differences in -5..5.

Original entry on oeis.org

1, 3, 7, 38, 187, 1237, 8235, 58524, 422504, 3117158, 23290361, 176103500, 1344136056, 10344670554, 80181862705, 625395356182, 4904914026691, 38658468142298, 306034794302017, 2432316965565820, 19401236375176220, 155259190622702818

Offset: 1

R. H. Hardin, Mar 04 2012

nonn

			Some solutions for n=6:
.-4...-3...-2...-3...-3...-3...-3...-3...-3...-3...-3...-3...-2...-2...-2...-3
.-1....0...-1....0...-2....1....0...-2....1...-2...-2...-3....0....1....0...-2
..0....1....0....0....1....2....3....2...-3....2....2...-2....1...-1....1....1
..0....0....3....0....4...-2....0...-1....1....2....1....1....2....1....1....1
..4....2...-1....1...-1....0...-2....2....2....3....0....5...-2....0...-2....3
..1....0....1....2....1....2....2....2....2...-2....2....2....1....1....2....0

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

Column 5 of A209032.

a(15)-a(22) from Andrew Howroyd, Mar 12 2017

A209030 Number of n-bead necklaces labeled with numbers -6..6 allowing reversal, with sum zero and first differences in -6..6.

Original entry on oeis.org

1, 4, 10, 60, 358, 2798, 22249, 186765, 1596162, 13907330, 122758794, 1096122478, 9880366438, 89796245827, 821927851286, 7570508171686, 70115797179348, 652594351781036, 6100765243003922, 57259681174637899, 539353919732122101, 5097036850039818860

Offset: 1

R. H. Hardin, Mar 04 2012

nonn

			Some solutions for n=6:
  -3  -6  -4  -4  -2  -3  -3  -3  -4  -6  -3  -3  -2  -4  -5  -5
  -1  -1  -3  -3   0  -2   1   0   1  -4  -2  -1  -1  -3   1  -5
   4  -1   0   2   1   2   3  -2  -3   2   2   4   2   2  -4   0
  -2   3   5   3  -1  -1  -1   1   0   3   2  -1   0   3   2   5
   0   5   4   4   0   1  -2   2   5   5   2  -1   0   0   5   4
   2   0  -2  -2   2   3   2   2   1   0  -1   2   1   2   1   1

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

Column 6 of A209032.

a(14)-a(22) from Andrew Howroyd, Mar 12 2017

A209031 Number of n-bead necklaces labeled with numbers -7..7 allowing reversal, with sum zero and first differences in -7..7.

Original entry on oeis.org

1, 4, 12, 88, 625, 5648, 52208, 505857, 4992048, 50169067, 510807173, 5260218485, 54684221779, 573168585795, 6050541798221, 64271927631233, 686512535921878, 7369066749454264, 79449368494683376, 859987831858141180, 9342319735394175920, 101820772577694786504

Offset: 1

R. H. Hardin, Mar 04 2012

nonn

			Some solutions for n=6:
.-6...-3...-5...-3...-2...-5...-4...-4...-3...-5...-4...-4...-5...-3...-3...-5
.-4...-1...-4...-1...-2...-2....1...-3....0...-3...-1...-1...-5...-1...-2...-2
..1...-2....2....5....0....2....0....1....1....4....2...-1....1...-1....4...-2
..4...-1....1...-2....0...-1....2....4...-2...-1....0....4....5....6....4....5
..6....5....6....0....2....4....0...-1....0....3....1....2....2...-1...-3....3
.-1....2....0....1....2....2....1....3....4....2....2....0....2....0....0....1

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

Column 7 of A209032.

a(14)-a(22) from Andrew Howroyd, Mar 12 2017

A209033 Number of 5-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first differences in -n..n.

Original entry on oeis.org

2, 11, 34, 88, 187, 358, 625, 1023, 1584, 2355, 3374, 4700, 6377, 8476, 11049, 14175, 17916, 22361, 27580, 33672, 40713, 48816, 58063, 68577, 80448, 93809, 108760, 125442, 143963, 164476, 187095, 211985, 239266, 269115, 301660, 337086, 375531

Offset: 1

R. H. Hardin, Mar 04 2012

nonn

Row 5 of A209032.

			Some solutions for n=6;
  -2 -5 -2 -5 -3 -4 -6 -4 -3 -1 -6 -3 -2 -4 -1 -2
   0 -1  0 -1  0  0 -1 -1  0 -1 -1 -1 -2  0  0 -2
  -1  0 -2  2  0  3  5  3 -3 -1  4  0  3 -1  0 -1
  -1  5  0  4  0  0  2  1  3 -1  4  4 -2  3  0  3
   4  1  4  0  3  1  0  1  3  4 -1  0  3  2  1  2

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A209032.

Empirical: a(n) = 2*a(n-1) - 2*a(n-3) + 2*a(n-4) - a(n-5) - 2*a(n-6) + 2*a(n-7) + a(n-8) - 2*a(n-9) + 2*a(n-10) - 2*a(n-12) + a(n-13).

Empirical g.f.: x*(2 + 7*x + 12*x^2 + 24*x^3 + 29*x^4 + 32*x^5 + 32*x^6 + 23*x^7 + 12*x^8 + 9*x^9 - x^11 + x^12) / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Jul 07 2018

A209034 Number of 6-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first differences in -n..n.

Original entry on oeis.org

4, 33, 144, 471, 1237, 2798, 5648, 10483, 18174, 29863, 46918, 71037, 104183, 148732, 207352, 283199, 379766, 501099, 651612, 836369, 1060813, 1331122, 1653894, 2036531, 2486878, 3013693, 3626176, 4334533, 5149447, 6082684, 7146500, 8354355

Offset: 1

R. H. Hardin, Mar 04 2012

nonn

Row 6 of A209032.

			Some solutions for n=6:
  -4  -2  -4  -5  -4  -3  -1  -2  -3  -2  -3  -4  -4  -6  -3  -3
  -2  -2  -2  -1  -1   1  -1  -1   0  -2  -2  -2  -3   0  -2  -2
   3   0   1   2   2   0   2   0  -1  -1  -2  -1   1  -1   4   1
   1  -2   4   1  -2   0  -1   3   3   1   1   4   5   1   0   4
   0   4  -1   2   4   0  -1  -2   0   1   3   1   0   6  -1   2
   2   2   2   1   1   2   2   2   1   3   3   2   1   0   2  -2

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = 2*a(n-2) + 2*a(n-3) - 3*a(n-5) - 3*a(n-6) - 2*a(n-7) + a(n-8) + 4*a(n-9) + 4*a(n-10) + a(n-11) - 2*a(n-12) - 3*a(n-13) - 3*a(n-14) + 2*a(n-16) + 2*a(n-17) - a(n-19).

A209035 Number of 7-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first differences in -n..n.

Original entry on oeis.org

6, 86, 576, 2517, 8235, 22249, 52208, 110285, 214440, 390344, 672932, 1108883, 1758433, 2698447, 4024458, 5854361, 8330650, 11624758, 15939472, 21513945, 28626249, 37599103, 48802532, 62660333, 79652814, 100324030, 125284570

Offset: 1

R. H. Hardin, Mar 04 2012

nonn

Row 7 of A209032.

			Some solutions for n=6:
  -5  -4  -5  -4  -2  -4  -3  -3  -5  -4  -3  -3  -3  -2  -6  -3
  -2  -3  -4  -1   0   1  -1  -2   0  -2   2   1   2  -1  -5  -2
   0   3  -1   0   1  -1   1   1  -2  -3  -1   1  -2   1  -1   3
   1   3   2   5  -2   2   1   2   3   0  -1  -2   3  -2   1   1
   4   0   4   1   1   1   2   1   0   5  -2  -2  -1  -1   5  -1
   4   2   3  -2   1  -1   1   3   3   4   3   3  -2   5   6   0
  -2  -1   1   1   1   2  -1  -2   1   0   2   2   3   0   0   2

R. H. Hardin, Table of n, a(n) for n = 1..144

Empirical: a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-7) - a(n-8) + a(n-10) + a(n-11) + 2*a(n-12) - 2*a(n-16) - a(n-17) - a(n-18) + a(n-20) + a(n-21) + a(n-23) - a(n-26) - a(n-27) + a(n-28).

cp's OEIS Frontend

A209025 Number of n-bead necklaces labeled with numbers -1..1 allowing reversal, with sum zero and first differences in -1..1.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A209026 Number of n-bead necklaces labeled with numbers -2..2 allowing reversal, with sum zero and first differences in -2..2.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A209027 Number of n-bead necklaces labeled with numbers -3..3 allowing reversal, with sum zero and first differences in -3..3.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A209028 Number of n-bead necklaces labeled with numbers -4..4 allowing reversal, with sum zero and first differences in -4..4.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A209029 Number of n-bead necklaces labeled with numbers -5..5 allowing reversal, with sum zero and first differences in -5..5.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A209030 Number of n-bead necklaces labeled with numbers -6..6 allowing reversal, with sum zero and first differences in -6..6.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A209031 Number of n-bead necklaces labeled with numbers -7..7 allowing reversal, with sum zero and first differences in -7..7.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A209033 Number of 5-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first differences in -n..n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A209034 Number of 6-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first differences in -n..n.

Views

Author

Keywords

Comments

Examples

Links

Formula

A209035 Number of 7-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first differences in -n..n.

Views

Author

Keywords

Comments

Examples