A052307 -id:A052307 - cp's OEIS frontend

A286110 Number of distinct hexaflexagons of length n.

1, 1, 1, 3, 3, 7, 8, 17, 21, 47, 63, 132, 205, 411, 685, 1353, 2385, 4643, 8496, 16430, 30735, 59343, 112531, 217245, 415628, 803209, 1545463, 2991191, 5778267, 11201883, 21702708, 42141575, 81830748, 159140895, 309590883, 602938098, 1174779397, 2290920127

Offset: 3

Michel Marcus, May 02 2017

nonn

Marshall Hampton, Constructing and Counting Hexaflexagons, arXiv:1704.07775 [math.CO], 2017.

Cf. A007282, A286111.

A286110 := proc(n)
    if type(n,'odd') then
        add(A052307(n,ceil(n/2)+1+3*i),i=0..n/6+1) ;
    else
        add(A052307(n,ceil(n/2)+3*i),i=0..n/6) ;
        %-A052307(n,n/2)/2+A007148(n/2)/2-1
    end if;
end proc:
seq(A286110(n),n=3..40) ; # R. J. Mathar, Jul 23 2017

A007148[n_] := (1/2)*(2^(n - 1) + Total[EulerPhi[2*#]*2^(n/#) & /@ Divisors[n]]/(2*n));
A052307[n_, k_] := Module[{hk = Mod[k, 2], a = 0}, If[k == 0, Return[1]]; Do[a = a + EulerPhi[d]*Binomial[n/d - 1, k/d - 1], {d, Divisors[GCD[k, n]]}]; (a/k + Binomial[Floor[(n - hk)/2], Floor[k/2]])/2];
a[n_] := Module[{s}, If[Mod[n, 2] == 1, Sum[A052307[n, Ceiling[n/2] + 1 + 3*i], {i, 0, Floor[n/6] + 1}], s = Sum[A052307[n, Ceiling[n/2] + 3*i], {i, 0, Floor[n/6] }]; s - A052307[n, n/2]/2 + A007148[n/2]/2 - 1]];
Table[a[n], {n, 3, 40}] (* Jean-François Alcover, Nov 28 2017, after R. J. Mathar *)

from sympy import binomial as C, totient, divisors, gcd, floor, ceiling
def a007148(n):
    if n==1: return 1
    return 2**(n - 2) + sum(totient(2*d)*2**(n//d) for d in divisors(n))//(4*n)
def a052307(n, k): return 1 if n==0 else C((n//2) - k%2 * (1 - n%2), (k//2))/2 + sum(totient(d)*C(n//d, k//d) for d in divisors(gcd(n, k)))/(2*n)
def a(n):
    if n%2: return sum([a052307(n, ceiling(n/2) + 1 + 3*i) for i in range(n//6 + 2)])
    else:
        s=sum([a052307(n, ceiling(n/2) + 3*i) for i in range(n//6 + 1)])
        return s - a052307(n, n//2)//2 + a007148(n//2)//2 - 1
print([a(n) for n in range(3, 41)]) # Indranil Ghosh, Jul 24 2017, after Maple code

A078925 Triangle of T1(n,m) = number of bracelets (necklaces that can be turned over) with m white beads and (2n+1-m) black ones, for 1<=m<=n.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 4, 7, 10, 1, 5, 10, 20, 26, 1, 6, 14, 35, 57, 76, 1, 7, 19, 56, 111, 185, 232, 1, 8, 24, 84, 196, 392, 600, 750, 1, 9, 30, 120, 324, 756, 1368, 2052, 2494, 1, 10, 37, 165, 507, 1353, 2829, 4950, 7105, 8524, 1, 11, 44, 220, 759, 2277, 5412, 10824

Offset: 1

Thomas Hartinger (hartinger_t(AT)web.de), Dec 15 2002

Left half of odd rows of table A052307 with left column deleted.

			1; 1, 2; 1, 3, 4; 1, 4, 7, 10; ...

Index entries for sequences related to bracelets

Cf. A052307 for full table, A073020 for even number of beads. Last term in each row gives A007123.

Table[ f[n, 2n + 1], {n, 11}] (* Robert G. Wilson v, Mar 29 2006 *)

A103441 Triangle read by rows: T(n,k) = number of bracelets of n beads (necklaces that can be flipped over) with exactly two colors and k white beads for which the set of distances among the white beads are different.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 3, 3, 1, 1, 3, 4, 4, 3, 1, 1, 4, 5, 7, 5, 4, 1, 1, 4, 7, 10, 10, 7, 4, 1, 1, 5, 8, 16, 13, 16, 8, 5, 1, 1, 5, 10, 20, 26, 26, 20, 10, 5, 1, 1, 6, 12, 28, 35, 35, 35, 28, 12, 6, 1, 1, 6, 14, 34, 57, 74, 74, 57, 34, 14, 6, 1, 1, 7, 16, 47, 73, 120, 85, 120, 73

Offset: 2

Wouter Meeussen, Feb 06 2005

If two bracelets can be made to coincide by rotation or flipping over they necessarily have the same set of distances, but the reverse is obviously not true.
Offset is 2, since exactly two colors are required, ergo at least two beads.
T[2n,n] equals A045611. Row sums equal A103442.
Same as A052307, except for bracelets such as {0,0,0,1,1,0,1,1} and{0,0,1,0,0,1,1,1}, that both have the same set of distances between the "1" beads: 4 d[0]+ 4 d[1]+ 2 d[2]+ 4 d[3]+ 2 d[4], where d[k] represents the unidirectional distance between two beads k places apart.

			Table starts as
  1;
  1,1;
  1,2,1;
  1,2,2,1;
  ...

Cf. A052307, A045611, A077078, A077079, A103442.

Needs[DiscreteMath`NewCombinatorica`]; f[bi_]:=DeleteCases[bi Range[Length[bi]], 0]; dist[li_, l_]:=Plus@@Flatten[Outer[d[Min[ #, l-# ]&@Mod[Abs[ #1-#2], l, 0]]&, li, li]]; Table[Length[Union[(dist[f[ #1], n]&)/@ListNecklaces[n, Join[1+0*Range[i], 0*Range[n-i]], Dihedral]]], {n, 2, 16}, {i, 1, n-1}]

A103691 Triangle read by rows: T(n,k) = number of bracelets of n beads (necklaces that can be flipped over) with exactly two colors and k white beads, for which the length (or abs value) of sum of the position vectors of the white beads are different.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 3, 3, 1, 1, 3, 4, 4, 3, 1, 1, 4, 4, 6, 4, 4, 1, 1, 4, 7, 10, 10, 7, 4, 1, 1, 5, 7, 11, 11, 11, 7, 5, 1, 1, 5, 10, 20, 26, 26, 20, 10, 5, 1, 1, 6, 10, 16, 18, 20, 18, 16, 10, 6, 1, 1, 6, 14, 34, 57, 74, 74, 57, 34, 14, 6, 1, 1, 7, 14, 33, 44, 53, 53, 53, 44, 33

Offset: 2

Wouter Meeussen, Feb 12 2005

Offset is 2, since exactly two colors are required, ergo at least two beads.

T[2n,n] equals A077078. Row sums equal A103692.

			T[8,3]=4 because of the 5 bracelets {1,1,1,0,0,0,0,0}, {0,0,0,0,1,0,1,1}, {0,0,0,1,0,0,1,1},{0,0,0,1,0,1,0,1} and {0,0,1,0,0,1,0,1}, the third and the fourth have equal absolute vector sums, length 1.
Table starts as:
  1;
  1,1;
  1,2,1;
  1,2,2,1;
  ...

Cf. A077078; A005648; A052307, A103692.

Needs[DiscreteMath`NewCombinatorica`]; f[bi_]:=DeleteCases[bi*Range[Length[bi]], 0]; vec[li_, l_]:= Abs[Plus@@ N[Exp[2*Pi*I*f[li]/l], 24]]; Table[Length[Union[(vec[ #, n]&)/@ ListNecklaces[n, Join[1+0*Range[i], 0*Range[n-i]], Dihedral], SameTest->(Abs[ #1-#2]<10^-18&)]], {n, 2, 16}, {i, 1, n-1}]

A211354 Refined triangle A211358: T(n,k) is the number of partitions up to rotation and reflection of an n-set that are of type k (k-th integer partition, defined by A194602).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 3, 1, 2, 1, 1, 3, 3, 8, 3, 6, 1, 5, 3, 3, 1, 1, 3, 4, 12, 4, 18, 3, 11, 9, 7, 1, 12, 3, 4, 1, 1, 4, 5, 22, 8, 38, 5, 39, 33, 25, 4, 57, 12, 19, 1, 17, 22, 25, 4, 5, 7, 1, 1, 4, 7, 30, 10, 76, 10, 85, 76, 55

Offset: 1

Tilman Piesk, Apr 09 2012

The rows are counted from 1, the columns from 0.
Row lengths: 1,2,3,5,7,11... (partition numbers A000041)
Row sums: 1,2,3,7,12,37... (A084708)
Row maxima: 1,1,1,2,3,8,18,57,228,668,3220
Distinct entries per row: 1,1,1,2,3,5,8,14,17,26,30
Rightmost columns are those from the triangle A052307  without the second column.

Tilman Piesk, Rows n=1..11 of triangle, flattened
Tilman Piesk, Partition related number triangles

Cf. A211358, A000041, A084708, A052307.

A211355 Refined triangle A211359: T(n,k) is the number of noncrossing partitions up to rotation and reflection of an n-set that are of type k (k-th integer partition, defined by A194602).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 3, 3, 5, 3, 3, 1, 2, 1, 1, 1, 1, 3, 4, 8, 4, 9, 3, 4, 4, 2, 1, 3, 1, 1, 1, 1, 4, 5, 14, 8, 19, 5, 14, 13, 8, 4, 12, 4, 4, 1, 3, 4, 3, 1, 1, 1, 1, 1, 4, 7, 20, 10, 38, 10, 30, 32, 16, 7, 48

Offset: 1

Tilman Piesk, Apr 09 2012

The rows are counted from 1, the columns from 0.
Row lengths: 1,2,3,5,7,11... (partition numbers A000041)
Row sums: 1,2,3,6,10,24... (A111275)
Row maxima: 1,1,1,2,2,5,9,19,48,132,330,781
Distinct entries per row: 1,1,1,2,2,4,6,9,15,21,28,43
Rightmost columns are those from the triangle A052307  without the second column.

Tilman Piesk, Rows n=1..12 of triangle, flattened
Tilman Piesk, Partition related number triangles

Cf. A211359, A000041, A111275, A052307.

cp's OEIS Frontend

A286110 Number of distinct hexaflexagons of length n.

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A078925 Triangle of T1(n,m) = number of bracelets (necklaces that can be turned over) with m white beads and (2n+1-m) black ones, for 1<=m<=n.

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A103441 Triangle read by rows: T(n,k) = number of bracelets of n beads (necklaces that can be flipped over) with exactly two colors and k white beads for which the set of distances among the white beads are different.

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A103691 Triangle read by rows: T(n,k) = number of bracelets of n beads (necklaces that can be flipped over) with exactly two colors and k white beads, for which the length (or abs value) of sum of the position vectors of the white beads are different.

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A211354 Refined triangle A211358: T(n,k) is the number of partitions up to rotation and reflection of an n-set that are of type k (k-th integer partition, defined by A194602).

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A211355 Refined triangle A211359: T(n,k) is the number of noncrossing partitions up to rotation and reflection of an n-set that are of type k (k-th integer partition, defined by A194602).

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