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).

1, 1, 5, 13, 42, 126, 387, 1180, 3606, 11012, 33636, 102733, 313781, 958384, 2927209, 8940617, 27307465, 83405605, 254747014, 778077690, 2376494563, 7258563604, 22169941574, 67713990832, 206819875428, 631693101321, 1929389878185

Offset: 0

Author

Emeric Deutsch, Apr 15 2005

Keywords

• nonn
• easy

Links

• E. Deutsch, Counting tilings with L-tiles and squares, Problem 10877, Amer. Math. Monthly, 110 (March 2003), 245-246.
• Index entries for linear recurrences with constant coefficients, signature (1,5,4,0,-1).

Programs

Maple

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);

Formula

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

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.