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A123520 Number of vertical dominoes in all possible tilings of a 2n X 3 grid by dominoes. 2
4, 28, 152, 744, 3436, 15284, 66224, 281424, 1178196, 4874444, 19973192, 81189688, 327817404, 1316035940, 5257118560, 20909651104, 82849544868, 327163551612, 1288036695544, 5057236343176, 19807689093644, 77408388584724 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000 (terms 0..100 from Vincenzo Librandi)

Index entries for linear recurrences with constant coefficients, signature (8,-18,8,-1).

FORMULA

a(n) = Sum_{k=0..n} 2^(k+1) * k * C(n+k,2*k).

a(n) = Sum_{k=0..n} k * A123519(n,k).

G.f.: 4*z*(1-z)/(1-4*z+z^2)^2.

a(n) = (2+sqrt(3))^n*((1+sqrt(3))*n+1/sqrt(3))/3 + (2-sqrt(3))^n*((1-sqrt(3))*n-1/sqrt(3))/3. - Vaclav Kotesovec, Nov 29 2012

EXAMPLE

a(1) = 4 because a 2 X 3 grid can be tiled in 3 ways with dominoes: 3 horizontal dominoes, 1 horizontal domino above two adjacent vertical dominoes and 1 horizontal domino below two adjacent vertical dominoes; these have altogether 4 vertical dominoes.

MAPLE

a:=n->sum(k*2^(k+1)*binomial(n+k, 2*k), k=0..n): seq(a(n), n=1..24);

MATHEMATICA

FullSimplify[Table[(2+Sqrt[3])^n*((1+Sqrt[3])*n+1/Sqrt[3])/3 + (2-Sqrt[3])^n*((1-Sqrt[3])*n-1/Sqrt[3])/3, {n, 1, 20}]] (* Vaclav Kotesovec, Nov 29 2012 *)

Table[Sum[2^(k + 1)*k*Binomial[n + k, 2 k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Oct 14 2017 *)

PROG

(PARI) z='z+O('z^50); Vec(4*z*(1-z)/(1-4*z+z^2)^2) \\ G. C. Greubel, Oct 14 2017

CROSSREFS

Cf. A001835, A123519.

Sequence in context: A006302 A272992 A272747 * A012847 A273431 A128721

Adjacent sequences:  A123517 A123518 A123519 * A123521 A123522 A123523

KEYWORD

nonn,easy

AUTHOR

Emeric Deutsch, Oct 16 2006

STATUS

approved

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Last modified October 6 15:48 EDT 2022. Contains 357269 sequences. (Running on oeis4.)