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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. 12
6, 30, 174, 1092, 7188, 48852, 339720, 2403588, 17236524, 124948668, 913820460, 6732898800, 49918950240, 372104853600, 2786716100592, 20955408717396, 158149624268220, 1197390368733804, 9091866006950892, 69214297980023256 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

L. Q. Eifler, K. B. Reid Jr., D. P. Roselle  "Sequences with adjacent elements unequal". Aequationes Mathematicae (1971), 6 (2-3), 256-262. doi:10.1007/BF01819761

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems.

FORMULA

a(n) = 2 *( Sum[k=0..[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)

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). - R. J. Mathar, Nov 01 2015

MATHEMATICA

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 *)

PROG

(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)+binomial(n-1, k+1)*binomial(2*n-2*k, n+1)))

CROSSREFS

Cf. A110707, A110710.

Sequence in context: A026331 A135490 A175925 * A001341 A089896 A057754

Adjacent sequences:  A110703 A110704 A110705 * A110707 A110708 A110709

KEYWORD

nonn

AUTHOR

Max Alekseyev, Aug 04 2005

STATUS

approved

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Last modified September 26 10:59 EDT 2017. Contains 292518 sequences.