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

0,2

Comments

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

Links

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.

Formula

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

Maple

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

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)))
(Magma) [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
(Sage) [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

Crossrefs

Cf. A110707, A110710.

Sequence in context: A175925 A362810 A365273 * A001341 A089896 A057754

Adjacent sequences: A110703 A110704 A110705 * A110707 A110708 A110709

Keyword

nonn

Author

Max Alekseyev, Aug 04 2005

Extensions

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

Status

approved