A105262 - OEIS

A105262

a(n)=number of tilings of a 4 X n rectangle using tiles that are either 1 X 1 squares or trominoes (here by a tromino we mean a 2 X 2 square with the upper right 1 X 1 square removed; no rotations allowed).

%I #9 Jul 26 2022 12:32:29

%S 1,1,5,13,42,126,387,1180,3606,11012,33636,102733,313781,958384,

%T 2927209,8940617,27307465,83405605,254747014,778077690,2376494563,

%U 7258563604,22169941574,67713990832,206819875428,631693101321,1929389878185

%N a(n)=number of tilings of a 4 X n rectangle using tiles that are either 1 X 1 squares or trominoes (here by a tromino we mean a 2 X 2 square with the upper right 1 X 1 square removed; no rotations allowed).

%H E. Deutsch, <a href="https://www.jstor.org/stable/3647950">Counting tilings with L-tiles and squares</a>, Problem 10877, Amer. Math. Monthly, 110 (March 2003), 245-246.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,4,0,-1).

%F G.f.: ( 1-x^2-x^3 ) / ( (1+x)*(x^4-x^3-3*x^2-2*x+1) ).

%F a(n) = a(n-1)+5a(n-2)+4a(n-3)-a(n-5) for n>=5; a(0)=1, a(1)=1, a(2)=5, a(3)=13, a(4)=42.

%p a[0]:=1:a[1]:=1:a[2]:=5:a[3]:=13:a[4]:=42: for n from 5 to 30 do a[n]:=a[n-1]+5*a[n-2]+4*a[n-3]-a[n-5] od: seq(a[n],n=0..30);

%K nonn,easy

%O 0,3

%A _Emeric Deutsch_, Apr 15 2005