A208597 -id:A208597 - cp's OEIS frontend

A208600 Number of 6-bead necklaces labeled with numbers -n..n not allowing reversal, with sum zero.

26, 297, 1564, 5457, 14838, 34153, 69784, 130401, 227314, 374825, 590580, 895921, 1316238, 1881321, 2625712, 3589057, 4816458, 6358825, 8273228, 10623249, 13479334, 16919145, 21027912, 25898785, 31633186, 38341161, 46141732, 55163249

Offset: 1

R. H. Hardin, Feb 29 2012

nonn

Row 6 of A208597.

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

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

Empirical: a(n) = (44/15)*n^5 + (22/3)*n^4 + (23/3)*n^3 + (14/3)*n^2 + (12/5)*n + 1.
Conjectures from Colin Barker, Mar 07 2018: (Start)
G.f.: x*(26 + 141*x + 172*x^2 + 8*x^3 + 6*x^4 - x^5) / (1 - x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.
(End)

A208601 Number of 7-bead necklaces labeled with numbers -n..n not allowing reversal, with sum zero.

Original entry on oeis.org

57, 1163, 8671, 39019, 129823, 353333, 833253, 1764925, 3438877, 6267735, 10816499, 17836183, 28300819, 43447825, 64821737, 94321305, 134249953, 187369603, 256957863, 346868579, 461595751, 606340813, 787083277, 1010654741

Offset: 1

R. H. Hardin, Feb 29 2012

nonn

Row 7 of A208597.

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

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

Cf. A208597.

Empirical: a(n) = (841/180)*n^6 + (841/60)*n^5 + (325/18)*n^4 + (51/4)*n^3 + (949/180)*n^2 + (37/30)*n + 1.
Conjectures from Colin Barker, Mar 07 2018: (Start)
G.f.: x*(57 + 764*x + 1727*x^2 + 750*x^3 + 71*x^4 - 6*x^5 + x^6) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
(End)

A208602 Number of n-bead necklaces labeled with numbers -1..1 not allowing reversal, with sum zero.

Original entry on oeis.org

1, 2, 3, 6, 11, 26, 57, 142, 351, 902, 2333, 6166, 16381, 44046, 119183, 324862, 890291, 2453126, 6789309, 18869426, 52635789, 147325510, 413618615, 1164517198, 3287073461, 9300516890, 26372968983, 74937177538, 213333642443, 608400919106, 1737954608281

Offset: 1

R. H. Hardin, Feb 29 2012

nonn

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

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

Column 1 of A208597.

comps[r_, m_, k_] := Sum[(-1)^i*Binomial[r - 1 - i*m, k - 1]*Binomial[k, i], {i, 0, Floor[(r - k)/m]}]; a[n_Integer, k_] := DivisorSum[n, EulerPhi[n/#] comps[#*(k + 1), 2 k + 1, #] &]/n; a[n_] = a[n, 1]; Array[a, 31] (* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)

a(n) = (1/n) * Sum_{d | n} totient(n/d) * A002426(d). - Andrew Howroyd, Mar 02 2017

cp's OEIS Frontend

A208600 Number of 6-bead necklaces labeled with numbers -n..n not allowing reversal, with sum zero.

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A208601 Number of 7-bead necklaces labeled with numbers -n..n not allowing reversal, with sum zero.

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A208602 Number of n-bead necklaces labeled with numbers -1..1 not allowing reversal, with sum zero.

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