A123520 Number of vertical dominoes in all possible tilings of a 2n X 3 grid by dominoes.

4, 28, 152, 744, 3436, 15284, 66224, 281424, 1178196, 4874444, 19973192, 81189688, 327817404, 1316035940, 5257118560, 20909651104, 82849544868, 327163551612, 1288036695544, 5057236343176, 19807689093644, 77408388584724

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