A110711 -id:A110711 - cp's OEIS frontend

A110707 Number of linear arrangements of n blue, n red and n green items such that there are no adjacent items of the same color (first and last elements considered as adjacent).

6, 24, 132, 804, 5196, 34872, 240288, 1688244, 12040188, 86892384, 633162360, 4650680640, 34390540320, 255773538240, 1911730760832, 14350853162676, 108139250403804, 817629606524112, 6200696697358344, 47152195812692664

Offset: 1

Max Alekseyev, Aug 04 2005

nonn

The number of linear arrangements is given by A110706 and the number of circular arrangements counted up to rotations is given by A110710.

Cf. A110706, A110710, A110711.

b = Binomial; a[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}]; Array[a, 20] (* Jean-François Alcover, Dec 04 2015, adapted from PARI *)

a(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)) ))

a(n) = 2 * Sum[k=0..[n/2]] binomial(n-1, k) * ( binomial(n-1, k)*(binomial(2n+1-2k, n+1)-3*binomial(2n-1-2k, n+1)) + binomial(n-1, k+1)*(binomial(2n-2k, n+1)-3*binomial(2n-2k-2, n+1)) )
a(n) = A110706(n) - A110711(n)
a(n) = 2*A000172(n-1)+2*A000172(n) - Mark van Hoeij, Jul 14 2010
Conjecture: n^2*a(n) -3*n*(2*n-1)*a(n-1) -3*(n-1)*(5*n-12)*a(n-2) -8*(n-3)^2*a(n-3)=0. - R. J. Mathar, Jul 26 2014
a(n) ~ 3^(3/2) * 2^(3*n - 1) / (Pi*n). - Vaclav Kotesovec, Nov 09 2024

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

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.

cp's OEIS Frontend

A110707 Number of linear arrangements of n blue, n red and n green items such that there are no adjacent items of the same color (first and last elements considered as adjacent).

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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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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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