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
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Oct 16 2006
STATUS
approved