A208970 -id:A208970 - cp's OEIS frontend

A208963 Number of n-bead necklaces labeled with numbers -1..1 allowing reversal, with sum zero and first and second differences in -1..1.

1, 1, 1, 1, 1, 2, 2, 4, 4, 7, 7, 14, 16, 32, 46, 90, 140, 258, 416, 746, 1245, 2228, 3845, 6929, 12194, 21990, 39113, 70544, 126293, 228168, 410887, 744377, 1346837, 2446229, 4441502, 8084433, 14717678, 26841848, 48977823, 89494630, 163634188, 299532409

Offset: 1

R. H. Hardin, Mar 03 2012

nonn

			All solutions for n=8:
.-1...-1....0...-1
.-1...-1....0...-1
..0....0....0...-1
..0....0....0....0
..1....0....0....1
..1....1....0....1
..0....1....0....1
..0....0....0....0

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

Column 1 of A208970.

a(34)-a(42) from Andrew Howroyd, Mar 12 2017

A208964 Number of n-bead necklaces labeled with numbers -2..2 allowing reversal, with sum zero and first and second differences in -2..2.

Original entry on oeis.org

1, 1, 1, 3, 3, 8, 15, 42, 94, 246, 613, 1645, 4361, 11980, 33023, 92484, 259762, 735310, 2088403, 5961138, 17070131, 49059542, 141402328, 408736264, 1184435227, 3440434897, 10014880408, 29211476668, 85362417183, 249882401380, 732670159161, 2151506490430

Offset: 1

R. H. Hardin, Mar 03 2012

nonn

			All solutions for n=5:
.-1...-1....0
..0...-1....0
..0....0....0
..1....1....0
..0....1....0

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

Column 2 of A208970.

a(23)-a(32) from Andrew Howroyd, Mar 12 2017

A208965 Number of n-bead necklaces labeled with numbers -3..3 allowing reversal, with sum zero and first and second differences in -3..3.

Original entry on oeis.org

1, 1, 2, 4, 9, 29, 87, 325, 1148, 4168, 15250, 57027, 214739, 820038, 3156251, 12243837, 47772129, 187370511, 738088612, 2919087297, 11585698838, 46133010366, 184243108172, 737834327019, 2962231908667, 11920345964795, 48072060624163, 194251242877839

Offset: 1

R. H. Hardin, Mar 03 2012

nonn

			All solutions for n=5:
..0...-2...-1...-1...-1...-2...-1...-1...-2
..0...-1...-1....0....0...-2...-1....0...-1
..0....1....0...-1....0....1....1....0....2
..0....2....1....1....1....2....0....0....2
..0....0....1....1....0....1....1....1...-1

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

Column 3 of A208970.

a(19)-a(28) from Andrew Howroyd, Mar 12 2017

A208966 Number of n-bead necklaces labeled with numbers -4..4 allowing reversal, with sum zero and first and second differences in -4..4.

Original entry on oeis.org

1, 2, 2, 8, 19, 90, 371, 1755, 8092, 37970, 179133, 857902, 4152036, 20313811, 100217735, 497983883, 2488970462, 12503041095, 63083893996, 319545209584, 1624399382535, 8284482761844, 42377064630483, 217362286106711, 1117715413419540, 5760869940768290

Offset: 1

R. H. Hardin, Mar 03 2012

nonn

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

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

Column 4 of A208970.

a(16)-a(26) from Andrew Howroyd, Mar 12 2017

A208971 Number of 4-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first and second differences in -n..n.

Original entry on oeis.org

1, 3, 4, 8, 11, 18, 24, 35, 45, 61, 76, 98, 119, 148, 176, 213, 249, 295, 340, 396, 451, 518, 584, 663, 741, 833, 924, 1030, 1135, 1256, 1376, 1513, 1649, 1803, 1956, 2128, 2299, 2490, 2680, 2891, 3101, 3333, 3564, 3818, 4071, 4348, 4624, 4925, 5225, 5551, 5876

Offset: 1

R. H. Hardin, Mar 03 2012

nonn

Row 4 of A208970.

			All solutions for n=5:
  -2   -1   -2   -2   -1   -1   -1    0   -2   -2   -1
   0   -1    0   -1    0    0    1    0   -1   -2   -1
   2    0    1    1    0    1   -1    0    2    2    1
   0    2    1    2    1    0    1    0    1    2    1

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

Cf. A208970.

Empirical: a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
Conjectures from Colin Barker, Jul 07 2018: (Start)
G.f.: x*(1 + x - 3*x^2 + x^3 + 2*x^4 - x^5) / ((1 - x)^4*(1 + x)^2).
a(n) = (2*n^3 + 6*n^2 + 28*n + 48) / 48 for n even.
a(n) = (2*n^3 + 6*n^2 + 22*n + 18) / 48 for n odd.
(End)

A208972 Number of 5-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first and second differences in -n..n.

Original entry on oeis.org

1, 3, 9, 19, 40, 77, 130, 213, 325, 484, 687, 956, 1294, 1715, 2233, 2863, 3612, 4508, 5551, 6779, 8186, 9814, 11667, 13773, 16153, 18832, 21824, 25171, 28874, 32991, 37513, 42502, 47963, 53938, 60455, 67549, 75241, 83587, 92589, 102325, 112783

Offset: 1

R. H. Hardin, Mar 03 2012

nonn

Row 5 of A208970.

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

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

Empirical: a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) + a(n-5) - 2*a(n-6) + 2*a(n-8) - a(n-9) + a(n-10) - 2*a(n-11) + 2*a(n-13) - a(n-14) - a(n-15) + 2*a(n-16) - 2*a(n-18) + a(n-19).

A208973 Number of 6-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first and second differences in -n..n.

Original entry on oeis.org

2, 8, 29, 90, 221, 495, 967, 1801, 3093, 5050, 7921, 11994, 17488, 25008, 34797, 47448, 63641, 83953, 108932, 139984, 177423, 222404, 276434, 340397, 415203, 503475, 605600, 723511, 859793, 1015565, 1192375, 1394565, 1622265, 1878352, 2167156

Offset: 1

R. H. Hardin, Mar 03 2012

nonn

Row 6 of A208970.

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

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

Empirical: a(n) = a(n-1) - a(n-2) + 3*a(n-3) - 2*a(n-4) + 2*a(n-5) - 3*a(n-6) + a(n-7) - a(n-8) + a(n-9) + a(n-10) - a(n-11) + 3*a(n-12) - 5*a(n-13) + 4*a(n-14) - 8*a(n-15) + 7*a(n-16) - 5*a(n-17) + 7*a(n-18) - 3*a(n-19) + 2*a(n-20) - 2*a(n-21) - 2*a(n-22) + 2*a(n-23) - 3*a(n-24) + 7*a(n-25) - 5*a(n-26) + 7*a(n-27) - 8*a(n-28) + 4*a(n-29) - 5*a(n-30) + 3*a(n-31) - a(n-32) + a(n-33) + a(n-34) - a(n-35) + a(n-36) - 3*a(n-37) + 2*a(n-38) - 2*a(n-39) + 3*a(n-40) - a(n-41) + a(n-42) - a(n-43).

A208974 Number of 7-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first and second differences in -n..n.

Original entry on oeis.org

2, 15, 87, 371, 1185, 3186, 7425, 15658, 30368, 55222, 95087, 156612, 248194, 380753, 567639, 825586, 1174510, 1638724, 2246594, 3032025, 4033918, 5298001, 6876080, 8828154, 11221516, 14133153, 17648642, 21864897, 26888831, 32840460, 39851308

Offset: 1

R. H. Hardin, Mar 03 2012

nonn

Row 7 of A208970.

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

R. H. Hardin, Table of n, a(n) for n = 1..79

A208962 Number of n-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first and second differences in -n..n.

Original entry on oeis.org

1, 1, 2, 8, 40, 495, 7425, 140429, 3014206, 73238115, 2000005946

Offset: 1

R. H. Hardin, Mar 03 2012

nonn

Diagonal of A208970.

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

A208967 Number of n-bead necklaces labeled with numbers -5..5 allowing reversal, with sum zero and first and second differences in -5..5.

Original entry on oeis.org

1, 2, 2, 11, 40, 221, 1185, 6883, 39143, 224991, 1299553, 7598452, 44893012, 267934513

Offset: 1

R. H. Hardin, Mar 03 2012

nonn

Column 5 of A208970.

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

cp's OEIS Frontend

A208963 Number of n-bead necklaces labeled with numbers -1..1 allowing reversal, with sum zero and first and second differences in -1..1.

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A208964 Number of n-bead necklaces labeled with numbers -2..2 allowing reversal, with sum zero and first and second differences in -2..2.

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A208965 Number of n-bead necklaces labeled with numbers -3..3 allowing reversal, with sum zero and first and second differences in -3..3.

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A208966 Number of n-bead necklaces labeled with numbers -4..4 allowing reversal, with sum zero and first and second differences in -4..4.

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A208971 Number of 4-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first and second differences in -n..n.

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A208972 Number of 5-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first and second differences in -n..n.

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A208973 Number of 6-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first and second differences in -n..n.

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A208974 Number of 7-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first and second differences in -n..n.

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A208962 Number of n-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first and second differences in -n..n.

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A208967 Number of n-bead necklaces labeled with numbers -5..5 allowing reversal, with sum zero and first and second differences in -5..5.

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