A110707 -id:A110707 - cp's OEIS frontend

A110706 Number of linear arrangements of n blue, n red and n green items such that there are no adjacent items of the same color.

1, 6, 30, 174, 1092, 7188, 48852, 339720, 2403588, 17236524, 124948668, 913820460, 6732898800, 49918950240, 372104853600, 2786716100592, 20955408717396, 158149624268220, 1197390368733804, 9091866006950892, 69214297980023256, 528150412279712856

Offset: 0

Max Alekseyev, Aug 04 2005

nonn

The number of circular arrangements is given by A110707 and A110710.

Seiichi Manyama, Table of n, a(n) for n = 0..1110 (terms 1..200 from Vincenzo Librandi)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems.
L. Q. Eifler, K. B. Reid Jr., D. P. Roselle, Sequences with adjacent elements unequal, Aequationes Mathematicae (1971), 6 (2-3), 256-262.

Cf. A110707, A110710.

[2*(&+[Binomial(n-1, k)*(Binomial(n-1, k)*Binomial(2*n+1-2*k, n+1) + Binomial(n-1, k+1)*Binomial(2*n-2*k, n+1)): k in [0..Floor(n/2)]]): n in [1..25]]; // G. C. Greubel, Nov 24 2018

a:= proc(n) option remember; `if`(n<2, 1+5*n,
      ((7*n-4)*a(n-1)+8*(n-2)^2*a(n-2)/(n+1))/n)
    end:
seq(a(n), n=0..21);  # Alois P. Heinz, Sep 09 2023

Table[2*(Sum[Binomial[n-1,k]*(Binomial[n-1,k]*Binomial[2n+1-2k, n+1]+Binomial[n-1,k+1]*Binomial[2n-2k,n+1]),{k,0,Floor[n/2]}]),{n,1,20}] (* Vaclav Kotesovec, Oct 18 2012 *)
Table[2 (Binomial[2 n + 1, n + 1] HypergeometricPFQ[{1 - n, 1 - n, 1/2 - n/2, -(n/2)}, {1, -(1/2) - n, -n}, 1] + (n - 1) Binomial[2 n, n + 1] HypergeometricPFQ[{1 - n, 2 - n, 1/2 - n/2, 1 - n/2}, {2, 1/2 - n, -n}, 1]), {n, 10}] (* Eric W. Weisstein, May 26 2017 *)
RecurrenceTable[{n(n+1)*a[n] == (n+1)*(7*n-4)*a[n-1] +8*(n-2)^2*a[n-2], a[1]==6, a[2]==30}, a, {n, 10}] (* Eric W. Weisstein, May 27 2017 *)

a(n)=2*sum(k=0,n\2,binomial(n-1,k)*(binomial(n-1,k)*binomial(2*n+1-2*k,n+1)+binomial(n-1,k+1)*binomial(2*n-2*k,n+1)))

[2*sum(binomial(n-1, k)*(binomial(n-1, k)*binomial(2*n+1-2*k, n+1) + binomial(n-1, k+1)*binomial(2*n-2*k, n+1))  for k in range(1+floor(n/2))) for n in (1..25)] # G. C. Greubel, Nov 24 2018

a(n) = 2 *( Sum_{k=0..floor(n/2)} binomial(n-1, k) * ( binomial(n-1, k) * binomial(2n+1-2k, n+1) + binomial(n-1, k+1)*binomial(2n-2k, n+1)) ).
a(n) = ((3*n-1)*A000172(n-1) + (3*n+2)*A000172(n))/(n+1).
D-finite with recurrence: n*(n+1)*a(n) = (n+1)*(7*n-4)*a(n-1) + 8*(n-2)^2*a(n-2). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 9*sqrt(3)*2^(3*n-2)/(Pi*n). - Vaclav Kotesovec, Oct 18 2012
G.f.: (2-x)*(1-8*x)^(-1/3)*(x+1)^(-2/3)*hypergeom([1/3, 1/3],[1],27*x^2/(8*x-1)/(x+1)^2) + 3*x*(2*x-1)^2*(1-8*x)^(-4/3)*(x+1)^(-8/3) * hypergeom([4/3, 4/3],[2],27*x^2/(8*x-1)/(x+1)^2) - 2. - Mark van Hoeij, May 14 2013
a(n) = 6*A190917(n) for n >= 1. - R. J. Mathar, Nov 01 2015

a(0)=1 prepended by Alois P. Heinz, Sep 09 2023

A110710 Number of ternary necklaces with n beads of each color and no adjacent beads of the same color (i.e., no substrings 00, 11, 22).

Original entry on oeis.org

1, 2, 5, 16, 70, 348, 1948, 11444, 70380, 445944, 2896590, 19186740, 129186596, 881808728, 6089851874, 42482906040, 298976142764, 2120377458900, 15141289233972, 108784152585236, 785869931659980, 5705406374249272

Offset: 0

Max Alekseyev, Aug 05 2005

nonn

The number of circular arrangements (counted up to rotations) of n blue, n red and n green items such that there are no adjacent items of the same color. The number of various linear arrangements is given by A110706, A110707 and A110711.

			For n=2 there are 5 necklaces: 010212, 012012, 012021, 012102, 021021.

G. C. Greubel, Table of n, a(n) for n = 0..1000
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Index entries for sequences related to necklaces

Cf. A000358, A110706, A110707, A110711.

b = Binomial; A110707[n_] := 2*Sum[b[n - 1, k]*(b[n - 1, k]*(b[2*n + 1 - 2*k, n + 1] - 3*b[2*n - 1 - 2*k, n + 1]) + b[n - 1, k + 1]*(b[2*n - 2*k, n + 1] - 3*b[2*n - 2*k - 2, n + 1])), {k,0, n/2}]; a[n_] := DivisorSum[n, A110707[n/#]*EulerPhi[#]&]/(3n); a[0]=1; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Dec 04 2015, adapted from PARI *)

{ A110707(n) = 2 * sum(k=0,n\2, binomial(n-1,k) * (binomial(n-1,k)*(binomial(2*n+1-2*k,n+1)-3*binomial(2*n-1-2*k,n+1)) + binomial(n-1,k+1)*(binomial(2*n-2*k,n+1)-3*binomial(2*n-2*k-2,n+1)) )); A110710(n) = sumdiv(n,d,A110707(n\d)*eulerphi(d))\(3*n); }

a(n) = Sum_{d|n} A110707(n/d)*eulerphi(d) / (3n) for n>0, a(0)=1.

a(n) ~ sqrt(3) * 2^(3*n - 1) / (Pi * n^2). - Vaclav Kotesovec, Mar 20 2023

a(0)=1 prepended by Alois P. Heinz, Dec 04 2015

A141147 Number of linear arrangements of n blue, n red and n green items such that the first item is blue and there are no adjacent items of the same color (first and last elements considered as adjacent).

Original entry on oeis.org

2, 8, 44, 268, 1732, 11624, 80096, 562748, 4013396, 28964128, 211054120, 1550226880, 11463513440, 85257846080, 637243586944, 4783617720892, 36046416801268, 272543202174704, 2066898899119448, 15717398604230888

Offset: 1

Max Alekseyev, Jun 10 2008

nonn

Max Alekseyev, PARI scripts for various problems
L. Q. Eifler, K. B. Reid Jr., and D. P. Roselle, Sequences with adjacent elements unequal, Aequationes Mathematicae 6 (2-3), 1971.

Cf. A110706, A110707, A110710, A110711, A141146, A141148.

A141147 := n -> 2^n*hypergeom([n, (1-n)/2, -n/2],[1, 1],1);
seq(simplify(A141147(i)),i=1..20); # Peter Luschny, Jan 15 2012

{ a(n) = sum(k=0,n\2, binomial(n,2*k) * binomial(2*k,k) * binomial(n-1+k,k) * 2^(n-2*k) ) }

a(n) = A110707(n) / 3 = (A000172(n) + A000172(n-1)) * 2 / 3.
a(n) = Sum_{k=0..floor(n/2)} binomial(n,2k) * binomial(2k,k) * binomial(n-1+k,k) * 2^(n-2k).
a(n) = 2^n*Hypergeometric([n,(1-n)/2,-n/2],[1, 1],1). - Peter Luschny, Jan 15 2012
Recurrence: (3*n^3 + 13*n^2 + 16*n + 4)*a(n+2) = (21*n^3 + 73*n^2 + 74*n + 16)*a(n+1) + (24*n^3 + 32*n^2)*a(n). - Ralf Stephan, Feb 11 2014
a(n) = (1/n) * Sum_{k = floor(n/2)..n} k * binomial(n,k)^2 * binomial(2*k,n). - Peter Bala, Mar 19 2023

A197657 Row sums of A194595.

Original entry on oeis.org

1, 4, 22, 134, 866, 5812, 40048, 281374, 2006698, 14482064, 105527060, 775113440, 5731756720, 42628923040, 318621793472, 2391808860446, 18023208400634, 136271601087352, 1033449449559724, 7858699302115444, 59906766929537116, 457685157123172664

Offset: 0

Susanne Wienand, Oct 17 2011

Number of meanders of length (n+1)*3 which are composed by arcs of equal length and a central angle of 120 degrees.
Definition of a meander:
A binary curve C is a triple (m, S, dir) such that
(a) S is a list with values in {L,R} which starts with an L,
(b) dir is a list of m different values, each value of S being allocated a value of dir,
(c) consecutive Ls increment the index of dir,
(d) consecutive Rs decrement the index of dir,
(e) the integer m>0 divides the length of S and
(f) C is a meander if each value of dir occurs length(S)/m times.
For this sequence, m = 3.
For 0 <= n <= 16, a(n) = the hypergraph Fuss-Catalan number FC_1^(2,n+1) in the notation of Chavan et al. - see 7.1 in the Appendix. - Peter Bala, Apr 11 2023

			Some examples of list S and allocated values of dir if n = 4:
Length(S) = (4+1)*3 = 15.
  S: L,L,L,L,L,L,L,L,L,L,L,L,L,L,L
dir: 1,2,0,1,2,0,1,2,0,1,2,0,1,2,0
  S: L,L,L,L,R,L,L,R,L,L,R,L,L,L,L
dir: 1,2,0,1,1,1,2,2,2,0,0,0,1,2,0
  S: L,R,L,L,L,L,L,R,L,L,R,L,R,R,R
dir: 1,1,1,2,0,1,2,2,2,0,0,0,0,2,1
Each value of dir occurs 15/3 = 5 times.

Parth Chavan, Andrew Lee and Karthik Seetharaman, Hypergraph Fuss-Catalan numbers, arXiv:2202.01111 [math.CO]
Peter Luschny, Meanders and walks on the circle.
Susanne Wienand, Animation of a meander
Susanne Wienand, Example of a meander

Cf. A198060, A198256, A198257, A198258.

A197657 := proc(n)
    (A000172(n) + A000172(n+1)) / 3 ;
end proc; # R. J. Mathar, Jul 26 2014
a := n -> 2^n*hypergeom([n + 1, -n/2, -n/2 - 1/2], [1, 1], 1):
seq(simplify(a(n)), n = 0..21); # Peter Luschny, Mar 26 2023

A197657[n_] := Sum[Sum[Sum[(-1)^(j + i)* Binomial[i, j]*Binomial[n, k]^3*(n + 1)^j*(k + 1)^(2 - j)/(k + 1)^2, {i, 0, 2}], {j, 0, 2}], {k, 0, n}]; Table[A197657[n], {n, 0, 16}]  (* Peter Luschny, Nov 02 2011 *)

A197657(n) = {sum(k=0,n,if(n == 1+2*k,3,(1+k)*(1-((n-k)/(1+k))^3)/(1+2*k-n))*binomial(n,k)^3)} \\ Peter Luschny, Nov 24 2011

def A197657(n):
    return 2^n*hypergeometric([n + 1, -n/2, -n/2 - 1/2], [1, 1], 1).simplify_hypergeometric()
for n in (0..21): print(A197657(n)) # Peter Luschny, Mar 26 2023

a(n) = Sum{k=0..n} Sum{j=0..2} Sum{i=0..2} (-1)^(j+i)*C(i,j)*C(n,k)^3*(n+1)^j*(k+1)^(2-j)/(k+1)^2. - Peter Luschny, Nov 02 2011
a(n) = Sum_{k=0..n} h(n,k)*binomial(n,k)^3, where h(n,k) = (1+k)*(1-((n-k)/(1+k))^3)/(1+2*k-n) if 1+2*k-n <> 0 else h(n,k) = 3. - Peter Luschny, Nov 24 2011
a(n) = A141147(n+1)/2 = A110707(n+1)/6 = (A000172(n)+A000172(n+1))/3. - Max Alekseyev, Jul 15 2014
Conjecture: (n+1)^2*a(n) -3*(n+1)*(2*n+1)*a(n-1) -3*n*(5*n-7)*a(n-2) -8*(n-2)^2*a(n-3)=0. - R. J. Mathar, Jul 26 2014
a(n) = 2^n*hypergeom([n + 1, -n/2, -n/2 - 1/2], [1, 1], 1). - Peter Luschny, Mar 26 2023
a(n) ~ sqrt(3) * 2^(3*n+1) / (Pi*n). - Vaclav Kotesovec, Apr 17 2023

A110711 Number of linear arrangements of n blue, n red and n green items such that first and last elements have the same color but there are no adjacent items of the same color.

Original entry on oeis.org

0, 6, 42, 288, 1992, 13980, 99432, 715344, 5196336, 38056284, 280658100, 2082218160, 15528409920, 116331315360, 874985339760, 6604555554720, 50010373864416, 379760762209692, 2891169309592548, 22062102167330592

Offset: 1

Max Alekseyev, Aug 04 2005

nonn

The number of linear arrangements is given by A110706 (first and last elements are not adjacent) and A110707 (first and last elements are adjacent) and the number of circular arrangements (counted up to rotations) is given by A110710.

Cf. A110706, A110707, A110710.

ogf := 6*((x-2)*hypergeom([1/3,1/3],[1], 27*x^2/((8*x-1)*(x+1)^2)) + 2*hypergeom([1/3,1/3],[2], 27*x^2/((8*x-1)*(x+1)^2))) / ((1-2* x)*(1+x)^(2/3)*(1-8*x)^(1/3));
series(ogf, x=0, 30); # Mark van Hoeij, Jan 22 2013

a(n) = 6 * sum(k=0,n\2, binomial(n-1,k) * ( binomial(n-1,k)*binomial(2*n-1-2*k,n+1) + binomial(n-1,k+1)*binomial(2*n-2*k-2,n+1) ))

a(n) = 6 * Sum_{k=0..floor(n/2)} binomial(n-1, k) * ( binomial(n-1, k)*binomial(2n-1-2k, n+1) + binomial(n-1, k+1)*binomial(2n-2k-2, n+1) ).
a(n) = A110706(n) - A110707(n).
a(n) = ((n-3)*A000172(n-1) + n*A000172(n))/(n+1). - Mark van Hoeij, Jul 14 2010
Conjecture: -(n+1)*(n-2)*a(n) + (7*n^2 - 13*n + 4)*a(n-1) + 8*(n-2)^2*a(n-2) = 0. - R. J. Mathar, Nov 01 2015

A141146 Number of linear arrangements of n blue, n red and n green items such that first and last elements are blue but there are no adjacent items of the same color.

Original entry on oeis.org

0, 2, 14, 96, 664, 4660, 33144, 238448, 1732112, 12685428, 93552700, 694072720, 5176136640, 38777105120, 291661779920, 2201518518240, 16670124621472, 126586920736564, 963723103197516, 7354034055776864, 56236603567496720

Offset: 1

Max Alekseyev, Jun 10 2008

nonn

Max Alekseyev, PARI scripts for various problems
L. Q. Eifler, K. B. Reid Jr., D. P. Roselle, Sequences with adjacent elements unequal, Aequationes Mathematicae 6 (2-3), 1971.

Cf. A110706, A110707, A110710, A110711, A141147, A141148.

{ a(n) = sum(k=0,n\2, binomial(n-1,2*k) * binomial(2*k,k) * binomial(n-1+k,k+1) * 2^(n-1-2*k) ) }

a(n) = A110711(n) / 3.
a(n) = Sum[k=0..[n/2]] binomial(n-1,2k) * binomial(2k,k) * binomial(n-1+k,k+1) * 2^(n-1-2k).
G.f.: (2*x-1)^2*(1-8*x)^(-4/3)*(x+1)^(-8/3)*hypergeom([4/3, 4/3],[2],27*x^2/((8*x-1)*(x+1)^2))-(1-8*x)^(-1/3)*(x+1)^(-2/3)*hypergeom([1/3, 1/3],[1],27*x^2/((8*x-1)*(x+1)^2)). - Mark van Hoeij, May 14 2013
Conjecture: -(n+1)*(n-2)*a(n) +(7*n^2-13*n+4)*a(n-1) +8*(n-2)^2*a(n-2)=0. - R. J. Mathar, Jul 23 2014

A141148 Number of aperiodic ternary necklaces with n beads of each color and no adjacent beads of the same color.

Original entry on oeis.org

2, 3, 14, 65, 346, 1929, 11442, 70310, 445928, 2896239, 19186738, 129184583, 881808726, 6089840427, 42482905678, 298976072384, 2120377458898, 15141288786096, 108784152585234, 785869928763325, 5705406374237814

Offset: 1

Max Alekseyev, Jun 10 2008

nonn

L. Q. Eifler, K. B. Reid Jr., D. P. Roselle, Sequences with adjacent elements unequal, Aequationes Mathematicae 6 (2-3), 1971.
Max Alekseyev, PARI scripts for various problems

Cf. A110706, A110707, A110710, A110711, A141146, A141147.

a(n) = Sum_{d|n} moebius(n/d) * A141147(d) / n.

A283614 T(n,k) = number of circular arrays of n 1's, n -1's, and k 0's such that no two adjacent elements are equal.

Original entry on oeis.org

1, 2, 6, 4, 2, 10, 24, 28, 12, 2, 14, 56, 132, 180, 132, 40, 2, 18, 100, 352, 804, 1196, 1120, 600, 140, 2, 22, 156, 728, 2324, 5196, 8160, 8840, 6300, 2660, 504, 2, 26, 224, 1300, 5320, 15844, 34872, 56848, 67900, 57820, 33264, 11592, 1848, 2, 30, 304, 2108, 10512, 39064, 110480, 240288, 402556, 515844, 496944, 348600

Offset: 0

Stefan Hollos, Apr 01 2017

The array is circular in the sense that the first and last elements are adjacent.

For linear arrays see A283613.

			The table starts with columns k=0..10 and rows n=0..5:
  | 0  1   2   3    4    5    6    7    8    9  10
-----------------------------------------------------------------
0 | 1
1 | 2  6   4
2 | 2 10  24  28   12
3 | 2 14  56 132  180  132   40
4 | 2 18 100 352  804 1196 1120  600  140
5 | 2 22 156 728 2324 5196 8160 8840 6300 2660 504
For n=2, k=3, the 28 arrays are:
[+0-0+0-] [+0+0-0-] [0-+0+0-] [0-0+0+-]
[0+-0+0-] [0+0-+0-] [0+0-0+-] [0+0+-0-]
[-0-0+0+] [-0+0-0+] [0-+0-0+] [0-0-+0+]
[0-0+-0+] [0-0+0-+] [0+-0-0+] [0+0-0-+]
[-+0-0+0] [-+0+0-0] [-0-+0+0] [-0+-0+0]
[-0+0-+0] [-0+0+-0] [+-0-0+0] [+-0+0-0]
[+0-+0-0] [+0-0-+0] [+0-0+-0] [+0+-0-0]

Cf. A110707, A265118, A283613.

nmax=8; Flatten[CoefficientList[Series[CoefficientList[Series[2*(x*y + 1)/Sqrt[(1 - y)*(1 - (2*x + 1)^2*y)] - 1, {y, 0, nmax }], y], {x, 0, 2nmax + 1 }], x]] (* Indranil Ghosh, Apr 02 2017 *)

G.f.: 2*(x*y+1)/sqrt((1-y)*(1-(2*x+1)^2*y))-1.
T(n,0)  G.f.: (1+y)/(1-y).
T(n,1)  G.f.: 2*y*(3-y)/(1-y)^2.
T(n,2)  G.f.: 4*y*(1+3*y-y^2)/(1-y)^3.
T(n,3)  G.f.: 4*y^2*(1+y)*(7-2*y)/(1-y)^4.
T(n,4)  G.f.: 4*y^2*(3+30*y+6*y^2-4*y^3)/(1-y)^5.
T(n,5)  G.f.: 4*y^3*(33+101*y-8*y^3)/(1-y)^6.
T(n,n) = A110707(n).
T(n,2*n) = 2*binomial(2*n,n).
Sum_{2*n+k = m} T(n,k) = A265118(m), m > 3.

cp's OEIS Frontend

A110706 Number of linear arrangements of n blue, n red and n green items such that there are no adjacent items of the same color.

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A110710 Number of ternary necklaces with n beads of each color and no adjacent beads of the same color (i.e., no substrings 00, 11, 22).

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A141147 Number of linear arrangements of n blue, n red and n green items such that the first item is blue and there are no adjacent items of the same color (first and last elements considered as adjacent).

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A197657 Row sums of A194595.

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A110711 Number of linear arrangements of n blue, n red and n green items such that first and last elements have the same color but there are no adjacent items of the same color.

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A141146 Number of linear arrangements of n blue, n red and n green items such that first and last elements are blue but there are no adjacent items of the same color.

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A141148 Number of aperiodic ternary necklaces with n beads of each color and no adjacent beads of the same color.

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A283614 T(n,k) = number of circular arrays of n 1's, n -1's, and k 0's such that no two adjacent elements are equal.