OFFSET
0,3
COMMENTS
Also the number of tilings of an n X 3 rectangle using integer-sided rectangular tiles of area n.
Also decimal expansion of 12443/109890 = 0.1132314132314... .
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,0,1,1).
FORMULA
G.f.: (-3*x^4-4*x^3-4*x^2-2*x-1) / (x^4+x^3-x-1).
From Wesley Ivan Hurt, Jun 20 2016: (Start)
a(n) + a(n-1) = a(n-3) + a(n-4) for n>4.
a(0) = 1, a(n) = (7 + 3*cos(n*Pi) + 2*cos(2*n*Pi/3))/3 for n>0. (End)
E.g.f.: 2*(-9/2 + cos(sqrt(3)*x/2)*exp(-x/2) + 2*sinh(x) + 5*cosh(x))/3. - Ilya Gutkovskiy, Jun 21 2016
EXAMPLE
a(6) = 4, because there are 4 tilings of a 6 X 3 rectangle using integer-sided rectangular tiles of area 6:
._._._. .___._. ._.___. ._____.
| | | | | | | | | | | |
| | | | | | | | | | |_____|
| | | | |___| | | |___| | |
| | | | | | | | | | |_____|
| | | | | | | | | | | |
|_|_|_| |___|_| |_|___| |_____|
MAPLE
a:=n-> `if`(n=0, 1, [4, 1, 3, 2, 3, 1][irem(n, 6)+1]): seq(a(n), n=0..100);
MATHEMATICA
PadRight[{1}, 120, {4, 1, 3, 2, 3, 1}] (* Harvey P. Dale, Jan 06 2016 *)
PROG
(Magma) [1] cat &cat [[1, 3, 2, 3, 1, 4]^^20]; // Wesley Ivan Hurt, Jun 20 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Alois P. Heinz, Dec 06 2012
STATUS
approved